Determinant Math Basic Concept

A determinant is a number associated with a matrix. Only SQUARE matrices have a determinant.

The symbol for a determinant can be the phrase “det” in front of a matrix variable, det(A); or vertical bars around a matrix, |A|

Minors and Cofactors:

If A is a square matrix, then the minor of entry aij is denoted by Mij and is defined to be the determinant of the sub-matrix that remains after the i number row and j number column are deleted from A.
The number (−1) i+j Mij is denoted by Cij and is called the cofactor of entry aij.

Determinant (Using cofactor expansion):

If A is an n × n matrix, then the number obtained by multiplying the entries in any row or column of A by the corresponding cofactors and adding the resulting products is called the determinant of A, and the sums themselves are called cofactor expansions of A. That is,

det(A) = a1 jC1 j + a2 jC2 j +… + anj Cnj [Cofactor expansion along the jth column] det(A) = ai1Ci1 + ai 2Ci 2 + … + ainCin [Cofactor expansion along the ith row]

Determinant of a triangular (upper or lower) matrix or diagonal matrix:

If A is an n × n triangular (upper triangular, lower triangular) or diagonal matrix, then det(A) is the product of the entries on the main diagonal of the matrix; that is, det(A) = a11a22 a33…ann .

Properties of Determinants of Matrices:

1. Determinant evaluated across any row or column is same.
2. If all the elements of a row or column are zeros, then the value of the determinant is zero.
3. Determinant of an Identity matrix  (In) is 1.
4. If rows and columns are interchanged then value of determinant remains same (value does not change). Therefore, det(A) = det(A^T), here A^T is transpose of matrix
5. If any two row or two column of a determinant are interchanged the value of the determinant is multiplied by -1.
6. If all elements of a row or column of a determinant are multiplied by some scalar number k, the value of the new determinant is k times of the given determinant. Therefore, if A be an n-rowed square matrix and K be any Then |KA| = Kⁿ|A| .
7. If two rows or columns of a determinant are identical (same) the value of the determinant is zero.
8. Let A and B be two matrix, then det(AB) = det(A)*det(B).
9. If A be a matrix then, |Aⁿ| = |A|ⁿ.
10. Determinant of Inverse of matrix can be defined as |A^-1| = |A|^-1.
11. Determinant of diagonal matrix, triangular matrix (upper triangular or lower triangular matrix) is product of element of the principal.
12. In a determinant each element in any row or column consists of the sum of two terms, then the determinant can be expressed as sum of two determinants of same For example, If B is obtained by adding c-times a row of A to a different row, the det(B) = det(A).
13. If value of determinant Δ becomes zero by substituting x = α, then x-α is a factor of α.
14. Here, cij denotes the cofactor of elements of aij in D.
15. In a determinant the sum of the product of the elements of any row (or column) with the cofactors of the corresponding elements of any other row or column is.. For example, det(A) = ai1Cj1 + ai2Cj2 + ai3Cj3 +…… + ainCjn, here Cj1, Cj2, Cj3 …Cjn are cofactors along elements of jth row.
16. Let λ1 , λ2 , λ3 ,…, λn are the Eigenvalues of A (square matrix of order n). Then det(A) = λ1λ2λ3 …λn , product of Eigenvalues.

DIFFERENCE BETWEEN MATRIX AND DETERMINANT:

Matrix	Determinant
A matrix is a rectangular arrangement of numbers, expressions, or symbols in rows and columns enclosed by ( ), [ ],  or || ||.	A determinant is a number associated with a matrix. It can be the phrase “det” in front of a matrix variable, det(A); or vertical bars around a matrix, |A|
A matrix cannot be reduced to a single number.	A determinant can be reduced to a single number
In a matrix, the number of rows may not be equal to the number of columns.	In a determinant, the number of rows must be equal to the number of columns.
An interchange of rows or columns gives a different matrix.	An interchange of rows or columns gives the same determinant with +ve or –ve sign.
Example:   ( 1    2 )	Example: |1|

Thank You!

Mechanics Physics Basic Concept

Classification of Motion:

1.  Based on duration: Periodic, Non periodic
2. Based on Speed: Uniform, non-uniform
3. Based on Path: Translatory, Circular, Rotatory, Oscillatory or Vibratory

Translatory Motion:

If a body moves as a whole such that all particles of the body moves with same velocity in straight parallel path, then the body is said to be in translatory motion.

Translatory motion can be of two types: Rectilinear and Curvilinear.

Rectilinear Motion:

When a body moves along a straight line  in such a way that each particle of the body travel the same distance , the motion is said to be a rectilinear motion.

Such ass: A car moving along a straight path and the train moving in a straight track.

Curvilinear Motion:

When the translator motion takes place  along a curved path then the motion is called curvilinear motion.

Such as: A stone thrown up in the air at a certain angle and a car taking a turn.

Rotational Motion:

Rotational motion refers to the movement of an object around an axis or center point. It involves spinning or turning about an imaginary line known as the axis of rotation.

Such as: Motion of an electric fan, motion of an analog clock.

Transla-Rotational/ Complex/ Mixed Motion:

When the motion of a body consists of both translation and rotation, then the motion is said to be transla-rotational motion.

Such as: Motion of the wheel of bicycle.

Periodic motion:

Any motion of a body that repeats itself after a regular interval of time is known as periodic motion. The time of repetition is called time period.

Such as: The earth moves once round the sun in 365 days. So the motion of the earth round the sun is periodic motion.

Similarly, hour or minute hand of a clock, piston of a cylinder in a car etc. are examples of periodic motion.

Vibrational motion:

When a body moves back and forth repeatedly about a mean position, its motion is called oscillatary or vibratory motion.

Such as: The pendulum of a wall-clock oscillates right and left about the mean or stable position.

So, motion of the pendulum of wall-clock is vibratory or oscillatory motion.

Circular Motion

Circular motion is a movement of an object along the circumference of a circle or rotation along a circular path.

Circular motion is one type of rotational motion and the axis through which the object rotates is called axis of rotation.

Such as: An artificial satellite orbiting the Earth at constant height, a stone which is tied to a rope and is being swung in circles, a car turning through a curve in a race track, an electron moving perpendicular to a uniform magnetic field

Angular displacement: θ = s/r

Consider a particle revolving around a point O in a circle of radius r. Let the position of the particle at time t = 0 be A and after time t, its position is B.

It can be measured by using a simple formula.

The formula is:  θ = s/r   | s = rθ     where,

θ is the angular displacement,

s is the distance travelled by the body, and

r is the radius of the circle along which it is moving.

In simpler words, the displacement of an object is the distance travelled by it around the circumference of a circle divided by its radius.

Angular velocity: ω = dθ/dt      ( ω = ω. + @t )

• The rate of change of angular displacement is called angular velocity.
• ω = dθ/dt

Acceleration In Circular Motion

No	Acceleration Type	Definition	Direction	Cause
1	Angular acceleration(α = dw/dt)	Measure of change in angular velocity.	Around the axis of rotation.	Caused by torque or force applied to rotating object.
2	Tangential accelerationat = rα=d|v|/dt	Acceleration along the tangent to the circular path.	Tangent to the circular path.	Caused by change in linear speed.
3	Centripetal accelerationac=v2/r=rw2	Acceleration directed towards the center of the circular path.	Towards the center of the circle.	Due to direction change of velocity.

People sometimes mix up angular and tangential (or linear) acceleration.

Angular acceleration is the change in angular velocity divided by time,

while tangential acceleration is the change in linear velocity divided by time.

People sometimes forget that angular acceleration

does not change with radius, but tangential acceleration does.

For example, for a rotating wheel that is speeding up, a point on the outside covers more distance in the same amount of time as a point closer to the center. It has a much larger tangential acceleration than the portion closer to the axis of rotation. However, the angular acceleration of every part of the wheel is the same because the entire object moves as a rigid body through the same angle in the same amount of time.

Linear Motion, Circular Motion, and Rotational Motion

Parameter	Linear Motion	Circular Motion	Rotational Motion
Definition	Motion along a straight path	Motion along a circular path	Motion around an axis or center point
Path	Straight line	Circular path	Depends on the object’s shape
Axis of Rotation	N/A	Center of the circle	Imaginary line passing through the center
Distance	Measured in meters (m)	Measured in meters (m)	Measured in radians or degrees
Velocity	Speed in a specific direction	Tangential speed along the circular path	Angular velocity around the axis
Acceleration	Change in velocity over time	Centripetal acceleration towards the center	Angular acceleration due to torque
Force	Linear force (F = m * a)	Centripetal force (Fc = m * v^2 / r)	Torque (τ = r × F)
Moment of Inertia	N/A	N/A	Measure of rotational inertia (I)
Conservation of Momentum	Linear momentum is conserved	Linear momentum is conserved	Angular momentum is conserved
Examples	Car moving along a road	Moon orbiting around the Earth	Spinning top, rotating Earth,
Examples	Train moving on a track	Satellite moving around a planet	Wheels of a car, etc.

Rigid Body:

When an external force acts on a body and the distance between the two points on the body doesn’t change, then the body is known as a Rigid Body.

Rigid body is a solid body with a fixed geometrical shape and size, both of which have negligible change during the motion or under the action of the applied forces.

The distance between any two given points on a rigid body remains constant in time regardless of external forces exerted on it.

The motion of a rigid body which is not pivoted or fixed in some way is either PURE TRANSLATION or a COMBINATION OF TRANSLATION AND ROTATION.

The motion of a rigid body which is pivoted or fixed in some way is rotation.

The rotation may be about an axis that is fixed (e.g. a ceiling fan) or moving (e.g. an oscillating table fan.).

We shall consider rotational motion about a fixed axis only.

Rotational motion:

Rotational motion refers to the movement of an object around an axis or center point. It involves spinning or turning about an imaginary line known as the axis of rotation.

Each particles of the object move in a circular path and the center of all circles lie on the axis of rotation.

Examples of rotational motion include a spinning top, the Earth rotating on its axis, the wheels of a car in motion, or a rotating fan.

Circular motion	Rotational motion
In circular motion, the object moves in a circular path around a fixed center, the center of rotation may be another object but not it’s own axis.	In rotational motion, the object rotates in circular path around an axis of rotation. The axis of rotation lies within itself.
The distance between the center of mass and the axis of rotation remains fixed.	Rotational movement is based on the rotation of the body around the center of mass.
The axis of rotation remains fixed.	The axis of rotation can change.
Artificial satellites, for example, orbit the Earth at a fixed altitude.	The Earth, for example, rotates on its own axis.

Parameter	Linear Motion	Symbol	Rotational Motion	Symbol
Description	Motion along a straight line.	–	Motion around a fixed axis.	–
Displacement	Change in position of an object.	s	Angle through which an object has rotated.	θ
Velocity	Rate of change of displacement.	v	Rate of change of angular displacement.	ω
Acceleration	Rate of change of velocity.	a	Rate of change of angular velocity.	α
Initial Velocity	Initial speed of the object.	u	Initial angular velocity.	ω0​
Time	Duration of motion.	t	Time of rotation.	t
Final Velocity	Velocity at a specific time.	v	Final angular velocity.	ω
Displacement at Time t	Position at a specific time.	s	Angle at time t.	θ
Kinematic Equations	Equations relating motion parameters.	–	Equations relating rotational parameters.	–
Initial Acceleration	Initial rate of change of velocity.	a	Initial angular acceleration.	α0​
Distance Traveled	Total path length covered.	d	Arc length traveled.	s
Force	Causes linear acceleration.	F	Torque causes angular acceleration.	τ
Mass	Measure of inertia in linear motion.	m	Moment of Inertia for rotation.	I
Newton’s Second Law	F=ma	–	Torque equation.	τ=Iα
Work	Force applied over a distance.	W	Work done in rotation.	W
Power	Rate of doing work.	P	Power in rotational motion.	P
Impulse	Change in momentum.	J	Change in angular momentum.	L
Conservation of Energy	Energy is conserved.	–	Conservation of angular momentum.	–

Center of Mass: Rcm = ( m1r1+m2r2…mnrn/m1+m2+…mn )

Center of mass of a system is the point that behaves as whole mass of the system is concentrated at it and all external forces are acting on it.

For rigid bodies, center of mass is independent of the state of the body that is whether it is in rest or in accelerated motion center of mass will remain same.

The interesting thing about the COM of an object or system is that it is the point where any uniform force on the object acts.

This is useful because it makes it easy to solve mechanics problems where we have to describe the motion of oddly-shaped objects and complicated systems.

Moment of Inertia: I = mr^2

The inertia of rotational motion is called moment of inertia. It is denoted by I. ( I = mr^2)

Moment of inertia is the property of an object by virtue of which it opposes any change in its state of rotation about an axis.

The moment of inertia of a body about a given axis is equal to the sum of the products of the masses of its constituent particles and the square of their respective distances from the axis of rotation.

The moment of inertia is only one of the numerous mass characteristics that may be used to quantify the stability of a structure as well as the amount of force required to change its motion.

When it comes to building construction, steadiness is an essential component that must be considered throughout the design
and production of various buildings.

Understanding the moment of inertia along different axes is crucial for evaluating a structure’s robustness against both external forces and internal forces.

In this article, we will go over all of the different features of the moment of inertia, as well as determine the moment of inertia of the disc.

What Is The Moment Of Inertia?

A body’s moment of inertia is equal to the product of the masses of all its particles multiplied by the square of their proximity from the rotation’s axis. Or, to put it another way,

it is the “quantity” that determines how much torque is required to achieve a certain angular acceleration around a rotating axis.

An object’s rotational inertia, or moment of inertia, is sometimes referred to as its angular mass.

When calculating moments of inertia, it is common practice to do so with reference to a particular axis of rotation.

The concentration of mass around a rotational axis is a primary factor in determining the outcome.

The moment of Inertia might be different depending on which axis is selected.

Why is the moment of inertia important?

The moment of inertia establishes how much torque is required for a given angular acceleration.
• Torque (or rotating force) is determined by the mass moment of inertia.
• The magnitude of torque needed to get a certain angular acceleration may be calculated by multiplying the moment of inertia by the angular acceleration. For a given acceleration, a higher moment of inertia number means more torque is needed.
• The designer’s ability to accurately identify these values is crucial for meeting the stringent performance requirements of the construction industry.
• The designer’s ability to strike the right balance between compactness, lightness, and efficiency is crucial to the success of any endeavour.
• Measuring MOI may also be used to ensure that the tolerances and targets of the production and assembly processes are acceptable.

Aspects Influencing Moment Of Inertia

• The following is a list of the fundamental elements that influence the moment of inertia:
• The density of the material
• The dimensions of the material
• The form that the substance takes
• Axis of rotation Physical significance of the moment of inertia
• The moment of inertia carries the same weight in terms of its physical implications as a mass that is moving in a linear direction.
• When determining a body’s inertia during translational motion, mass is the most important factor to consider.
• The magnitude of an object’s moment of inertia grows as its mass does.
• The force that is necessary to produce linear acceleration will, as a result, increase.
• When anything is moving in a rotating motion, the angular acceleration will be higher if the moment of inertia is larger.

Radius of Gyration: K = Square{(r1^2+r2^2+…rn^2)/n}

Radius of gyration or Gyradius of a body about an axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body’s actual distribution of mass, if the total mass of the body were concentrated.

It is the imaginary radius from the reference axis where the whole mass is assumed to be concentrated.

Torque: τ = r x F = rF sinθ n

Torque is the twisting effect of the force applied to a rotating object which is at a position r from its axis of rotation.

Mathematically, this relationship is represented as follows: τ = r x F = rF sinθ n

Torque τ is defined as a quantity in rotational motion, which when multiplied by a small angular displacement gives us work done in rotational motion. This quantity corresponds to force in linear motion, which when multiplied by a small linear displacement gives us work done in linear motion.

PHYSICAL SIGNIFICANCE OF TORQUE:

Torque in rotational motion is same as force in linear motion. It does is include the angular rotation. Otherwise torque is the force that would cause displacement.

Torque is the turning effect of a force about the axis of rotation.
τ = r x F = rF sinθ n
It is a vector quantity. If the nature of the force is to rotate the object clockwise, then torque is called negative and if rotate the object anticlockwise, then it is called positive.
Its SI unit is ‘newton-metre’ and its dimension is [ML2T-2].

In rotational motion, torque, τ = Iα where a is angular acceleration and 1is moment of inertia.
• Torque is the twisting effect of the force applied to a rotating object which is at a position r from its axis of rotation. Mathematically,
this relationship is represented as follows:

• Physicists often discuss torque within the context of equilibrium, even though an object experiencing net torque is definitely not in
equilibrium.
• Balance and Equilibrium: Torque is essential in determining the rotational equilibrium of an object.
• For an object to be in equilibrium, the sum of the torques acting on it must be zero. This is known as the “principle of moments.”
• In practical terms, this means that an object will remain at rest or rotate at a constant angular velocity if the total torque acting on it
is balanced.
• This is why a balanced seesaw or a balanced bicycle wheel stays stationary or rotates steadily.
• In fact, torque provides a convenient means for testing and measuring the degree of rotational or circular acceleration experienced by an object, just as other means can be used to
calculate the amount of linear acceleration.
• In equilibrium, the net sum of all forces acting on an object should be zero; thus in order to meet the standards of equilibrium, the sum of all torques on the object should also be zero.

Applications of Torque

Rotational Systems:

Torque is fundamental to understanding and designing rotational systems like gears, pulleys, and flywheels. It’s necessary for determining the force required to start or stop these systems and to maintain their rotation.

Engines and Motors: Torque is essential for engines and motors, such as internal
combustion engines and electric motors. In the case of engines, torque is used to
calculate the power output, and in motors, torque is a key parameter for
specifying their performance.

Tightening Bolts: Torque wrenches are used to apply a specific torque to bolts
during assembly. This ensures that bolts are properly tightened without causing
damage due to over-tightening.

• Robotic Arm Movement: Torque is used to control the movement of
robotic arms and joints. Servo motors apply torque to achieve precise and
controlled motion.

Industrial Machinery: In automated manufacturing processes, torque is
employed to control the movement of conveyor belts, robotic assembly
arms, and other machinery.

• Angular Acceleration Studies: Torque is used in experiments to study
rotational motion and angular acceleration, allowing researchers to
investigate properties of materials and objects.

Conservation of Angular Momentum: Torque is central to the conservation
of angular momentum, a principle that explains how the total angular
momentum of a closed system remains constant unless acted upon by an
external torque.

• Vehicle Performance: Torque plays a crucial role in determining
a vehicle’s acceleration and towing capacity. High torque at low
speeds contributes to better acceleration, while high torque
throughout the RPM range ensures efficient operation.

• Transmission Systems: Torque converters and clutches in
automatic and manual transmissions manage the transfer of
torque from the engine to the wheels, enabling smooth gear
changes and optimal power delivery.

• Spacecraft Attitude Control: Torque is used to control the
orientation (attitude) of spacecraft. Reaction wheels, which spin
to generate torque, help adjust the spacecraft’s position without
expelling mass (conservation of angular momentum).

• Gyroscope Systems: Gyroscopes utilize torque to maintain their
angular orientation, providing stabilization and navigation
assistance for spacecraft and aircraft.

Orthopedics:

Understanding the torque applied to joints and bones helps orthopedic surgeons design implants and prosthetics that can withstand the forces experienced during movement.

Angular momentum: L = I ω      ( I = mr )

Angular momentum, property characterizing the rotary inertia of an object or system of objects in motion about an axis that may or may not pass through the object or system.

The Earth has orbital angular momentum by reason of its annual revolution about the Sun and spin angular momentum because of its daily rotation about its axis.

The moment of linear momentum is called angular momentum. It is denoted by L.

• Linear momentum is a product of the mass (m) of an object and the velocity (v) of the object. If an object has higher momentum, then it harder to stop it.
• The formula for linear momentum is p = mv.
• The total amount of momentum never changes, and this property is called conservation of momentum. Let us study more about Linear momentum and conservation of momentum.

Angular momentum and linear momentum are examples of the parallels
between linear and rotational motion.
They have the same form and are subject to the fundamental constraints
of conservation laws, the conservation of momentum and the
conservation of angular momentum.

Principle of Conservation of Angular Momentum

If no external torque acts on a rotating body then angular momentum of the body remains constant.

We know that,
Example of principle of conservation of angular momentum:
• Ballet dancers rotate themselves about their feet as axis with constant angular momentum. When they fold their hands near the body, the M.I. decreases, and the angular velocity increases and they rotates fast. But, if they stretch their arms away from the body, the M.I. increases and angular velocity decreases and they rotate slow.

MOMENT OF INERTIA OF A CIRCULAR DISC

Consider a uniform circular disc of mass M and radius R, rotating about an axis passing through its centre and perpendicular to its
plane.
Now, the area of the disc, A = π r 2
Mass of the disc = M Mass per unit area of the disc, = M π R
Let us consider, a small circular strip of width dx at a distance x from the centre of the disc. The area of the strip dA = Circumference of the strip x width of the strip

dA = 2 x × dx If dm is the mass of the strip, then
dm = × 2 x dx dm
dm = M π R 2 × 2 x dx
dm = 2M R 2 x dx

Why is the moment of inertia calculated?

In terms of how it affects motion in a straight line, the moment of inertia plays the same part as mass.
A body’s rotational inertia is the amount of force it takes to alter its direction of rotation. It remains the same for each given rigid frame and any given rotational axis. When compared to a circular disc, why does a ring have a larger moment of inertia?
When compared to a circular disc of the same radius and mass, a ring’s moment of inertia is larger along an axis
that passes through its centre of mass and is transverse to its plane. As a result of its mass being concentrated at its outer edge, furthest from its central axis, a ring has a greater moment of inertia.

Summary of moment of inertia
• In physics, moment of inertia is a crucial concept that
describes how an object’s mass is distributed relative to
its rotational axis. It plays a significant role in
understanding rotational motion and is particularly
important in the study of classical mechanics and rigid
body dynamics.
• In summary, the moment of inertia is a fundamental
property that governs rotational motion and helps us
understand how objects behave when subjected to
rotational forces. Its physical significance lies in
describing rotational inertia, angular acceleration,
conservation of angular momentum, rigid body rotation,
and gyroscopic effects.

Rotational Motion:

Rotational motion refers to the movement of an object around an axis or center point. It involves spinning or turning about an imaginary line known as the axis of rotation.

Examples of rotational motion include a spinning top, the Earth rotating on its axis, the wheels of a car in motion, or a rotating fan.

Circular Motion:

Circular motion is the motion of an object along a circular path, but not necessarily around an axis.
This motion can occur horizontally, vertically, or at an angle.
Examples include the moon orbiting the Earth, a satellite moving around a planet, or a ball attached to a string being swung in a circle.

Axis of Rotation:

The axis of rotation is an imaginary line that passes through the center of rotation.

any point on the object that lies on this line remains stationary during the rotation.

For example, in the case of a spinning top, the axis of rotation runs through its central point, allowing it to spin around this axis.

Angular Displacement:

Angular displacement is the change in the object’s orientation or angle relative to the axis of rotation.

It is measured in degrees or radians and indicates how far the object has rotated from its initial position.

Relationship between Linear and Angular Motion

Linear Displacement vs Angular Displacement:

Linear displacement refers to the change in position of an object along a straight line.
Angular displacement, as mentioned earlier, is the change in the object’s orientation with respect to the axis of rotation.

The relationship between the two can be understood through trigonometry.

Linear Velocity vs. Angular Velocity:

Linear velocity is the rate of change of linear displacement,

while angular velocity is the rate of change of angular displacement.

Both velocities are related through the radius of the circular path and can be calculated using specific formulas.

Tangential Speed:

Tangential speed is the linear speed of an object at any given point on its circular path.

It is always perpendicular to the radius and is equal to the product of the angular velocity and the radius.

Torque in Rotational Motion:

Torque is the turning force that causes an object to rotate around an axis.

It is similar to force in linear motion but acts in a rotational sense. Torque is required to change the rotational state of an object or cause angular acceleration.

Moment Arm:

The moment arm (lever arm) is the perpendicular distance from the axis of rotation to the line of action of the force.

It determines the effectiveness of the force in producing rotational motion.

Torque Formula:

The torque acting on an object is given by the formula τ = r × F, where τ is the torque, r is the moment arm, and F is the applied force. Torque is measured in Newton-meters (Nm) or foot pounds (Ft-lbs).

Moment of Inertia:

The moment of inertia is a property of an object that describes its resistance to changes in its rotational motion.

It depends on the mass distribution of the object and the axis of rotation. Objects with larger moment of inertia are more resistant to changes in their rotational state.

Moment of Inertia Formula:

For a collection of particles or rigid bodies, the moment of inertia (I) is calculated as the sum of each particle’s mass (m) multiplied by the square of its distance (r) from the axis of rotation, summed up for all particles. Mathematically, I = Σ(mr2).

Centripetal Force in Circular Motion:

Centripetal force is the force that acts on an object moving in a circular path, keeping it on its circular trajectory.

This force always points towards the center of the circle.

Centripetal Force Formula:

For an object of mass m moving at a constant speed v along a circular path of radius r, the centripetal force (Fc ) required to maintain the circular motion is given by the formula Fc = (mv2 ) / r.

Centrifugal Force (Misconception)

Centrifugal force is often misunderstood as an actual force acting outward on an object in circular motion.

However, it is a fictitious force that appears to act outward when an observer is in a non-inertial (accelerating) reference frame. It is a result of inertia rather than a real force.

Apparent Force:

To an observer in a rotating frame of reference, the centrifugal force seems to counteract the centripetal force, making it appear as if objects are pushed outward from the center of rotation. Conservation of Angular Momentum

Law of Conservation:

The angular momentum of a closed system remains constant unless acted upon by an external torque.

This principle is analogous to the law of conservation of linear momentum in linear motion.

Angular Momentum Formula:

The angular momentum (L) of an object rotating with an angular velocity (ω) and moment of inertia (I) is given by L = I * ω. When no external torques act on the system, the angular momentum remains constant.

Short Question from Inertia

1. Question: Why does a ball roll off the table if you give it a push?
Answer: When you push a ball on the table, it wants to keep moving in the same direction because of inertia. Inertia is
when things want to keep doing what they’re already doing. So, the ball keeps rolling off the table because it wants to
keep moving.
2. Question: How come it’s hard to stop a bike once it’s moving?
Answer: It’s hard to stop a bike once it’s moving because of inertia. When the bike is going, it wants to keep going
forward. You have to use the brakes to slow it down because the bike naturally wants to keep moving forward.
3. Question: Why does it feel like you’re being pushed back when a car speeds up quickly?
Answer: When a car speeds up quickly, your body wants to stay in one place because of inertia. So, when the car moves
forward, your body feels like it’s being pushed back because it wants to stay still.
4. Question: Why do things fall off a spinning merry-go-round when it slows down?
Answer: When a merry-go-round slows down, things want to keep moving because of inertia. So, things on the merrygo-round fly off because they want to keep moving in a straight line, even though the merry-go-round is slowing down.
5. Question: How come it’s easier to keep pushing a toy car than to start pushing it from a stop?
Answer: It’s easier to keep pushing a toy car because of inertia. Once the car is moving, it wants to keep moving
because of inertia. Starting to push it from a stop is harder because you have to overcome its resistance to getting
moving in the first place.
Short Question from FORCE
1. If you were a superhero with the power to control force, how would you use it to help people?
I would use my force-controlling powers to stop disasters like earthquakes or floods by stabilizing buildings or redirecting
floodwaters away from populated areas.
2. Imagine you’re designing a new amusement park ride based on the concept of force. What would it look like and how
would it work?
The ride would be a giant swing that uses powerful magnets to create forces that push and pull riders in different directions,
giving them a thrilling experience of changing gravitational forces.
3. If you could change one thing about how force works in the universe, what would it be and why?
I would make it possible for humans to control gravitational force, allowing us to levitate objects effortlessly or create artificial
gravity in space, making space travel much easier and safer.
4. How do you think animals use force in nature, and can we learn anything from them?
Answer: Animals use force in various ways, such as birds using air resistance to glide, or predators using forceful movements
to catch their prey. We can learn from them by studying their techniques and applying them to our own inventions and
technologies.
5. If you were stranded on a deserted island and needed to build a shelter using only the forces available in nature, how
would you do it?
I would use the force of gravity to stack large rocks together to form walls, and then use vines or branches to create a roof that
can withstand the forces of wind and rain.
6. How do you think force could be used in art or creative expression?
Force could be used to create kinetic sculptures that move in response to external forces, or in interactive installations where
visitors can experience and manipulate different forces.
7. If you could invent a new force that doesn’t exist yet, what would it be and what would it do?
I would invent a force called “harmony force” that brings objects and people closer together in peaceful and cooperative ways,
promoting understanding and unity among all beings.
8. Imagine a world where force behaves differently than it does in our universe. How would everyday life be different?
In this world, objects might float instead of falling to the ground, or pushing something might make it move faster
instead of slowing it down. Everyday tasks like walking or driving would require new strategies and techniques.
9. If you could communicate with force as if it were a sentient being, what questions would you ask it?
I would ask force how it feels to be such a fundamental aspect of the universe and what it thinks about the ways that
humans harness and manipulate it for various purposes.
10. How do you think force could be used in futuristic technology or inventions that haven’t been created yet?
Force could be used to create advanced propulsion systems for spacecraft, or in medical devices that use controlled
forces to manipulate cells and tissues for healing and regeneration.
11. What is friction?
Answer: Friction is a force that opposes motion between two surfaces that are in contact with each other. It can make
it harder for objects to slide past each other.
12. How does air resistance affect objects?
Answer: Air resistance is a type of friction that acts on objects as they move through the air. It can slow down objects
like parachutes or skydivers.
13. What is gravitational force?
Answer: Gravitational force is the force of attraction between two objects with mass. It’s what keeps us on the ground
and causes objects to fall towards the Earth.
14. How does force affect the motion of objects?
Answer: Force can change the speed, direction, or shape of an object. It can make objects speed up, slow down, or
change their path of motion. Without force, objects would remain at rest or continue moving in a straight line at a
constant speed.
15. What are some examples of forces in everyday life?
Answer: Examples of forces include pushing a door open, pulling a wagon, kicking a ball, or gravity pulling objects
towards the ground.
16. How do we measure force?
Answer: Force is measured in units called Newtons (N). We use tools like spring scales or force meters to measure
how much force is being applied to an object.
Short Question from NEWTONS THIRD LAW
1. Question: Why does a balloon zoom away when you blow air into it?
Answer: When you blow air into a balloon, you’re pushing air out of your mouth into the balloon. At the same time,
the balloon pushes back on the air. This is Newton’s Third Law: for every action, there is an equal and opposite reaction.
So, the air pushes the balloon forward, making it zoom away!
2. Question: How do rockets go up into the sky?
Answer: Rockets have powerful engines that shoot hot gases out of the bottom. When the gases shoot down, the rocket
pushes up against them. At the same time, the gases push back on the rocket. This makes the rocket go up into the sky,
just like a balloon goes up when you let the air out.
3. Question: Why do you move backward when you push against a wall?
Answer: When you push against a wall, you’re pushing on it with your hands. The wall pushes back on you with the
same force. This is Newton’s Third Law. So, when you push on the wall, you also get pushed backward a little bit.
4. Question: How do boats move forward in the water?
Answer: Boats have propellers that spin really fast. When the propellers spin, they push water backward. At the same
time, the water pushes back on the propellers with the same force. This makes the boat move forward through the
water.
5. Question: Why do you feel a kick when you jump off a diving board?
Answer: When you jump off a diving board, your feet push against it to push you up into the air. At the same time, the
diving board pushes back on your feet. This is Newton’s Third Law. So, you feel a kick as the diving board pushes
back, sending you flying into the air!
Short Question from IMPULSE
1. If you could harness the power of impulse in your daily life, how would you use it to achieve your goals?
Answer: I would use impulse to take quick, decisive actions towards my goals without overthinking or hesitating,
seizing opportunities as they arise.
2. Imagine you have the ability to manipulate impulse in others. How would you use this power to influence positive
change in the world?
Answer: I would use my power to inspire people to act on their impulses for kindness, generosity, and social justice,
fostering a world where spontaneous acts of goodness are commonplace.
3. If you were to design a game based on the concept of impulse, what would it look like and how would it be played?
Answer: The game would involve quick decision-making and rapid response challenges, where players must act on
their impulses to navigate obstacles and achieve objectives within a limited time frame.
4. How do you think understanding impulse could improve communication and relationships between people?
Answer: Understanding impulse can help people empathize with each other’s spontaneous reactions and behaviors,
fostering better communication and more authentic connections in relationships.
5. If you could travel back in time and change one impulsive decision you made, what decision would it be and why?
Answer: I would change the impulsive decision to quit a hobby or activity that I later regretted, allowing myself to
experience the joy and fulfillment it brought me in the long run.
6. How do you think impulse plays a role in creativity and artistic expression?
Answer: Impulse can drive artists to follow their instincts and explore new ideas without inhibition, leading to
spontaneous and innovative works of art that captivate and inspire audiences.
7. If you could create a device that measures and visualizes impulse, how would it work and what insights do you think
it would reveal about human behavior?
Answer: The device would analyze physiological and neurological signals to quantify impulses in real-time, providing
valuable insights into the subconscious motivations and emotional states driving human behavior.
8. Imagine a world where people had perfect control over their impulses. How would society be different?
Answer: In such a world, people might exhibit greater self-discipline and restraint in their actions, leading to fewer
conflicts, better decision-making, and more harmonious relationships within communities.
9. How do you think impulse could be harnessed in education to enhance learning experiences for students?
Answer: Impulse could be harnessed to create dynamic and engaging learning environments that encourage curiosity,
exploration, and hands-on experimentation, fostering a deeper understanding and retention of knowledge.
10. If you could write a story where impulse was the central theme, what would the plot be and what lessons would it
convey?
Answer: The story could revolve around a character who learns to navigate the consequences of their impulsive
decisions, ultimately discovering the importance of balance, self-awareness, and mindful reflection in shaping their
destiny.
Short Question from MOMENTUM
1. If you possessed the ability to control momentum, how would you use it in your everyday life?
Answer: I would use momentum to accomplish tasks more efficiently, such as giving myself a running start to jump
higher or pushing objects with greater force while expending less effort.
2. Question: How does a football player run faster using momentum?
Answer: A football player runs faster by taking big steps and building up speed. When they run, they have momentum,
which helps them keep going forward. It’s like when you push a toy car and it rolls far away.
3. Question: Why does a swinging pendulum keep moving back and forth?
Answer: A swinging pendulum keeps moving because of its momentum. When you push it to one side, it has
momentum that carries it to the other side. It swings back and forth until the momentum runs out.
4. Question: How do roller coasters use momentum to make fun rides?
Answer: Roller coasters use momentum to zoom around twists and turns. They start by going up a big hill, which gives
them lots of momentum to speed down and do loops and curves. It’s like a fast train ride that makes your tummy tickle!
5. Question: Why do baseballs hit by a bat go far?
Answer: When a baseball is hit by a bat, it gains momentum from the force of the swing. This momentum helps it fly
through the air. The harder the hit, the more momentum the ball gets, so it goes farther.
6. Question: How does a bouncing ball show momentum in action?
Answer: When you bounce a ball, it goes up and down because of its momentum. When you throw it down, it gains
momentum and bounces back up. It keeps bouncing until the momentum slows down, like a super springy toy!
7. Imagine a world where momentum worked differently than it does in our universe. How would people adapt to this
unique property of motion?
Answer: In this world, people might need to adjust their movements and strategies to account for unpredictable changes
in momentum, leading to innovative techniques for transportation, sports, and daily activities.
8. If you were to design a roller coaster based on the principles of momentum, what features would it have and how would
it thrill riders?
Answer: The roller coaster would feature steep drops, sharp turns, and loops designed to maximize changes in
momentum and create exhilarating sensations of acceleration and weightlessness for riders.
9. How do you think understanding momentum could revolutionize transportation systems of the future?
Answer: Understanding momentum could lead to the development of more efficient and sustainable transportation
systems, such as high-speed trains or hyperloop technologies, that harness momentum to propel vehicles at incredible
speeds with minimal energy consumption.
10. If you could create a machine that amplifies momentum, what practical applications could it have in various industries?
Answer: The machine could be used to launch payloads into space with greater efficiency, propel vehicles with
incredible acceleration, or generate renewable energy by harnessing the kinetic energy of moving objects.
11. How do you think momentum influences the dynamics of teamwork and collaboration in group activities?
Answer: Momentum can build within a team as members work together towards a common goal, propelling them
forward with a sense of shared momentum and momentum can also build within a team as members work together
towards a common goal, propelling them forward with a sense of shared purpose and momentum.
12. If you could manipulate momentum to create a new form of entertainment, what would it be and how would it captivate
audiences?
Answer: I would create a gravity-defying performance art show where performers use momentum to execute
breathtaking stunts and acrobatics, defying the laws of physics and leaving audiences in awe.
13. Imagine a sport where momentum played a central role in gameplay. How would it be played and what strategies would
athletes employ to gain an advantage?
Answer: The sport could involve teams competing to maintain control of a momentum-generating object, such as a
ball or puck, while strategically harnessing momentum to outmaneuver opponents and score points.
14. How do you think momentum could be used to create innovative solutions for environmental challenges, such as
renewable energy generation or waste management?
Answer: Momentum could be harnessed to drive turbines and generators for renewable energy production, or to propel
vehicles and machinery powered by sustainable sources like wind, water, or solar energy.
15. If you could explore a fictional world where momentum was the governing force of nature, what wonders and dangers
would you encounter?
Answer: In this world, landscapes might be characterized by towering cliffs and cascading waterfalls where momentum
dictates the flow of rivers and the behavior of natural phenomena, presenting both exhilarating opportunities for
exploration and perilous challenges for adventurers to overcome.
Short Question from ANGULAR ACCELERATION
1. Question: Why does a spinning top speed up when you pull the string?
Answer: When you pull the string of a spinning top, you’re giving it more force to spin faster. This makes it change its
speed quickly, and that change in speed is called angular acceleration. So, the top speeds up because of angular
acceleration.
2. Question: How come a figure skater spins faster when they pull their arms in close to their body?
Answer: When a figure skater pulls their arms in close to their body while spinning, they’re changing how fast they’re
spinning. This change in spinning speed is called angular acceleration. So, they spin faster because of angular
acceleration.
3. Question: Why does a bicycle wheel slow down when you put your hand against it?
Answer: When you put your hand against a spinning bicycle wheel, you’re giving it a force to slow down. This makes
it change its speed quickly, and that change in speed is called angular acceleration. So, the wheel slows down because
of angular acceleration.
4. Question: How does a tornado start spinning so fast?
Answer: A tornado starts spinning fast because of angular acceleration. As the air starts swirling around, it changes its
spinning speed quickly, and that’s angular acceleration. This makes the tornado spin faster and faster.
5. Question: Why does a record player speed up when you turn the knob?
Answer: When you turn the knob on a record player, you’re giving it a force to spin faster. This makes it change its
spinning speed quickly, and that change in speed is called angular acceleration. So, the record player speeds up because
of angular acceleration.
Short Question from TORQUE
1. Why is it easier to open a tight jar lid with a longer spoon? | A longer spoon creates more torque by increasing the
perpendicular distance between your force and the lid. More torque equals easier turning!
2. How does a screwdriver tighten a screw? | Torque generated by twisting the handle applies a rotational force to the
screw, causing it to rotate and dig into the material.
3. Why do car tires need to be inflated? | Inflated tires have a larger radius of gyration (mass further from the center).
This reduces inertia and makes turning easier, requiring less torque.
4. Why are doorknobs round? | Round knobs are easier to grip and apply torque in any direction, unlike lever handles that
require specific hand positioning.
17. How does a seesaw work? | People’s weight applies torque in opposite directions. The side with greater torque (more
weight or further from the fulcrum) goes down!
Short Question from RADIUS OF GYRATION
1. Why is a tennis ball easier to spin than a basketball?
Tennis ball has a smaller radius of gyration (mass closer to center), meaning less inertia to overcome and start spinning.
2. Why do ice skaters spin faster with arms tucked in?
Tucking arms closer reduces radius of gyration, making them more compact and easier to spin with the same force.
3. Why are fan blades wider near the edges?
Wider blades have a larger radius of gyration, creating more angular momentum as they spin, leading to stronger air
circulation.
4. Why are pendulum clocks with slower ticks longer?
Longer pendulum has a larger radius of gyration, resulting in a longer oscillation period (slower tick) for the clock.
5. Why are baseball bats thicker at the end?
Thicker end has a larger radius of gyration, increasing moment of inertia. This helps transfer more energy to the ball
when hitting, resulting in a stronger swing.
6. Imagine you’re playing with spinning tops. Why do the bigger ones spin slower than the smaller ones?
Think of the spinning top like a ballerina. When the top is bigger, it’s like the ballerina stretching her arms out wide.
This increases the radius of gyration, making it harder to spin fast because it has more “stuff” moving further away
from the center. The smaller top, like a ballerina with arms tucked in, has a smaller radius of gyration and spins faster
with the same push.
7. Why do ice skaters spin faster when they pull their arms in?
It’s like the spinning top again! Imagine the ice skater with arms outstretched, like the ballerina. Pulling their arms in
makes them smaller, similar to tucking the arms in the top. This reduces the radius of gyration, making them spin faster
without needing more energy.
8. Why are bicycle wheels round?
Round wheels have the same radius of gyration at every point, making them spin smoothly and efficiently. If they were
shaped differently, different parts of the wheel would have different distances from the center, making the spin wobbly
and uneven.
9. Why is it easier to open a tight jar lid with a long spoon?
The longer spoon acts like a lever, increasing the perpendicular distance between your hand and the lid. This creates a
bigger torque (twisting force) with the same amount of pushing, like using a longer wrench to loosen a bolt. It’s all
about using distance to your advantage!
10. Why do baseball bats have thicker ends than handles?
The thicker end has a larger radius of gyration when swung. This increases the moment of inertia, meaning it takes
more energy to slow down. When you hit the ball, this extra inertia helps transfer more energy from your swing to the
ball, making it fly farther and faster!
Short Question from MOMENT OF INERTIA
1. Question: How does a figure skater use moment of inertia to spin beautifully on the ice?
Answer: When a figure skater pulls their arms in close to their body while spinning, they decrease their moment of
inertia. This makes it easier for them to spin faster and perform graceful spins.
2. Question: Why are heavy objects harder to move and stop?
Answer: Heavy objects have more moment of inertia, which means they resist changes in their motion. That’s why it’s
harder to get them moving or to stop them once they’re in motion.
3. Question: How does a gymnast tuck their body to spin faster during a flip?
Answer: When a gymnast tucks their body during a flip, they decrease their moment of inertia. This allows them to
spin faster and perform flips more quickly and smoothly.
4. Question: Why does a car’s engine need more power to accelerate when it’s fully loaded with passengers and luggage?
Answer: When a car is fully loaded, it has more mass, which increases its moment of inertia. This means the engine
needs more power to overcome the increased resistance to acceleration
.
5. Question: How does a diver adjust their body position to execute perfect twists and turns during a dive?
Answer: A diver adjusts their body position to change their moment of inertia. By tucking or extending their body, they
can control their rotation speed and perform precise twists and turns in the air.

Thank You! 

Dynamics Physics Basic Concept

Mechanics

Mechanics is the branch of physics that deals with the study of motion, forces, and the behavior of physical systems.

It forms the foundation of classical physics and plays a crucial role in understanding various natural phenomena and engineering applications.

Rigid Body

When an external force acts on a body and the distance between the two points on the body doesn’t change, then the body is known as a Rigid Body.

Mechanics is the branch of physics which describes and predicts the conditions of rest or motion of bodies under the action of forces.

In a word, it is the branch of Physics dealing with the study of static body and its motion.

It turns out that everything that happens in the world is some type of motion.

Motion of blood in our body, motion of our eye, signal from brain, hand movement, all over motion,
students should be able to solve problems related to Newtons laws of motion, projectile motion, dynamics, circular and rotational motion, moment of inertia, linear momentum and angular momentum, moment,
torque, impulse etc.

Newton’s Laws of Motion

Isaac Newton (a 17th century scientist) put forth a variety of laws that explain why objects move (or don’t move) as they do. These three laws have become known as Newton’s three laws of motion.

Newton’s First Law of Motion:

Newton’s first law states that ‘an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force’.

The tendency of a body to remain at rest or in uniform linear motion is called inertia,
and Newton’s first law is often called the law of inertia.
Basically what Newton’s First Law is saying is that objects behave predictably. If a
ball is sitting on your table, it isn’t going to start rolling or fall off the table unless a
force acts upon it to cause it to do so. Moving objects don’t change their direction
unless a force causes them to move from their path.

Newton’s Second Law of Motion:

Newton’s second law states that ‘The rate of change of momentum of a body is directly proportional to the impressed force and takes place in the direction of the force.’

When a force acts on an object, it will cause the object to accelerate. The larger the
mass of the object, the greater the force will need to be to cause it to accelerate.
This Law may be written as, force = mass x acceleration
or: F = m a
If you apply more force to an object, it accelerates at a higher rate. If the same force
is applied to an object with greater mass, the object accelerates at a slower rate
because mass adds inertia.

Newton’s Third Law of Motion:

Newton’s third Law of Motion states that ‘for every action, there is an equal and opposite reaction’.

When one body exerts a force on a second body, the second body simultaneously
exerts a force equal in magnitude and opposite in direction on the first body.

Force:

Force is a kind of impact, external or internal which tends to change or in real sense.

it changes the state of rest or uniform motion of a body in a straight line i. e. it
changes the inertia of any object.
Forces can be used to increase the speed of an object, decrease the speed of an object,
or change the direction in which an object is moving.
Force is a vector quantity. The unit of force is Newton (N).

Momentum
The momentum of a particle is defined as the product of its mass times its velocity.
The linear momentum is a vector quantity, possessing a direction as well as a
magnitude,
Momentum = mass x velocity
P = mv
It’s direction along the direction of velocity. The SI unit for momentum is kg m/s.
Linear momentum is also a conserved quantity, meaning that if a closed system is
not affected by external forces, its total linear momentum cannot change.

Galileo’s Laws of Motion

Galileo (1564-1642) was the first to determine, at the start of the seventeenth century, the law of constant acceleration of free-falling bodies. Galileo gave three laws about falling bodies. These are called Galileo’s laws in the case of falling bodies. These Laws are applicable to freely falling bodies. The law states that the distances traveled are proportional to the squares of the elapsed times.

Laws are given below:

First law: In a vacuum, all the freely falling bodies starting from rest traverse equal distance at equal interval of time or, in a vacuum all bodies starting from rest fall with equal rapidity.

Explanation: According to this law if different objects having different sizes, namely volumes are allowed to fall from rest at a particular height and without any resistance will reach the ground at the same time. The first law can be proved by the guinea and feather experiment.

Second law: Starling from rest, the velocity of a freely falling body is proportional to the time taken to fall. If a falling body gets velocity v at time t, it can be mathematically written as, v ∞ t.

Explanation: If a body is allowed to fall freely from rest under the action of gravitational force, then if its velocity in one second is v, then the velocity in two seconds will be 2v, in three seconds 3v and so on. Generally, if the velocities of the falling body in t₁, t₂, t₃ secs, are respectively v₁, v₂, v₃ then we find,

v₁/t₁ = v₂/t₂ = v₃/t₃

So, v ∞ t

Third law: Starting from rest, the distance traversed by a freely falling body is proportional to the square of the time of fall. If a falling body traverses distance h in time t, it can be mathematically written as, h ∞ t².

Explanation: Under gravitational attraction, if a body, initially at rest, is allowed to fall freely then the body will traverse distance h in 1 sec., distance 2² x h in 2 sec., 3² x h in 3 sec. and so on. In other words, in equal successive periods of time, the distances traveled by a free-falling body are proportional to the succession of odd numbers (1, 3, 5, 7, etc.). So if the body traverses distances h₁, h_2, and h respectively, in t₁, t₂, t₃ secs., then,

h₁/t₁² = h₂/t₂² = h₃/t₃²

so, h ∞ t².

Software engineers may need physics because it helps with

• logical thinking,
• debugging issues,
• solving complex problems, and
• working with physics-related applications.

Having a deep understanding of physics won’t make you an amazing software engineer. However, it can help develop certain abilities that will help you improve.

Motion: Describes how objects move in space and time. It involves concepts like displacement, velocity, and acceleration.
Forces: Influences that cause changes in motion. Newton’s laws of motion govern the relationship between forces and motion.
Energy: The ability to do work or cause changes in a system. It exists in different forms, such as kinetic, potential, and mechanical
energy.
Momentum: A property of moving objects that depends on their mass and velocity. Conservation of momentum is an essential principle in mechanics.

Equilibrium and Statics:
Equilibrium: Occurs when them net force and net torque acting on an object are both zero. Objects in equilibrium do not accelerate.
Statics: The study of systems in equilibrium, where forces are balanced and there is no motion.
Kinematics: Describes the motion of objects without considering the forces causing it.
Involves concepts like displacement, velocity, acceleration, and time. Formulas for uniformly accelerated motion are crucial in
solving kinematic problems.

Dynamics: Focuses on understanding the forces that cause motion and their effects on objects.
Involves the application of Newton’s laws of motion to analyze real-world scenarios.
Work: The product of force and displacement along the direction of the force.
Energy: The capacity to do work. Mechanical energy is the sum of kinetic and potential energy.
Power: The rate at which work is done or energy is transferred.

What is a Rigid Body?
When an external force acts on a body and the distance between the two points on the body doesn’t change, then the body is known as a Rigid Body.
Or it can be said that a body that does not change shape under the influence of forces is known as a Rigid Body.

CONCEPT OF A POINT MASS OR PARTICLE

If a particle is moving along a curved path in a plane, then it is said to be in two dimensional motion.

MOTION CLASSIFICATION: DIMENSION CLASSIFICATION OF MOTION SCALAR

Scalar quantities are the Physical quantities
which is complete with a magnitude alone. Mass, Temperature, Energy, Work, Pressure

The mass of my body is 60 kg means my body is made up of matter 60kg. It doesn’t need any more explanation.

VECTOR

Vector quantity is the one which need a direction for completing its existence.
Velocity, Force/ Weight, Acceleration.

My weight is 600 N means, I am pulled towards earth with a force of 600N. It is not complete unless it is specified towards Earth.

Parameters of Motion
Position
Distance
Displacement
Speed (average, instantaneous)
Velocity (average, instantaneous)
Acceleration (average, instantaneous)

Distance & Displacement

Displacement is the shortest distance i.e. the difference in position of the object. It is the straight line
distance between the initial and final positions of an object. It is a vector quantity.

Distance is the length of the path followed by the object. It is a scalar quantity.

DIFFERENCE BETWEEN DISTANCE & DISPLACEMENT

Distance ≥ Displacement
Example: Calculate the displacement vector
for a particle moving from a point P to Q as
shown below. Calculate the magnitude of
displacement.

Example: An athlete covers 3 rounds on a circular track of radius 50 m. Calculate the total distance and displacement
travelled by him.

Solution: The total distance the athlete covered =3x circumference of track
The displacement is zero, since the athlete reaches the same point A after three rounds from where he started.

SPEED
Speed is the rate of change of distance or the distance covered per unit time
• Speed is the total distance
(s) covered in total time (t)

Differentiate between reference point a frame of reference. Locate a point (-3, -4) in a reference frame.

Reference Point:

• A reference point is a specific, identifiable location used as a starting point to describe the position or motion of objects.
• Nature: It is a singular point chosen as a baseline for measurements.
• Role: It serves as a point of comparison for determining the location of other objects.
• Example: In a Cartesian coordinate system, the origin (0, 0) is a common reference point.

Frame of Reference:

• Definition: A frame of reference is a coordinate system used to specify the location or motion of objects relative to a set of axes.
• Nature: It is a broader concept that includes multiple points, axes, and a coordinate system.
• Role: It provides a spatial context for describing the position, velocity, and acceleration of objects.
• Example: The coordinate system in which you describe the motion of a car on a road.

to locate the point (-3, -4) in a reference frame:

• A Cartesian coordinate system with an x-axis and a y-axis.
• Start from the reference point (0, 0), which is the origin.
• Move 3 units to the left along the x-axis (negative direction) from the origin.
• Then, move 4 units downward along the y-axis (negative direction) from the x-axis position reached.
• The resulting point (-3, -4) is now located in the reference frame.

A reference point is a specific location used as a starting point for measurements, while a frame of reference is a coordinate system providing a spatial context. The point (-3, -4) is located by moving from the reference point within a reference frame.

What is the condition for circular motion to exist?

Circular motion exists when an object travels along a circular path with a constant speed, experiencing a centripetal force directed toward the center of the circle. The object’s inertia and centripetal acceleration maintain this motion, requiring sufficient tangential velocity and a central force field.

Circular motion exists when an object moves along a circular path, maintaining a constant distance from a fixed point known as the center of the circle. For circular motion to occur, certain conditions must be met:

1. **Constant Speed:** The object must move at a constant speed. While the speed remains constant, the direction of motion continuously changes, leading to circular motion.

2. **Centripetal Force:** There must be a centripetal force acting on the object, directing it towards the center of the circular path. This force is necessary to keep the object in circular motion and counteract its tendency to move in a straight line.

3. **Central Force Field:** The centripetal force is often provided by a central force field, such as gravitational force or tension in a string. This force is always directed toward the center of the circular path.

4. **Inertia and Centripetal Acceleration:** The object’s inertia, wanting to move in a straight line, is overcome by the centripetal force, causing it to continuously change direction. The acceleration of the object is directed toward the center of the circle and is called centripetal acceleration.

5. **Sufficient Tangential Velocity:** The object must have sufficient tangential velocity, which is the component of velocity tangent to the circular path. This velocity, combined with the centripetal force, maintains the circular motion.

These conditions collectively ensure that an object moves in a circular path rather than a straight line. It’s important to note that if any of these conditions is not met, the circular motion may not be sustained.

Define force. Characteristics of a force? Derive the impulse momentum formula.

**Force:**
Force is a push or pull with both magnitude and direction.

**Characteristics:**
1. Magnitude (strength).
2. Direction (line of action).
3. Point of application.
4. Contact or action-at-a-distance nature.

**Impulse-Momentum Formula Derivation:**
Impulse (\( \Delta p \)) is equal to force (\( F \)) multiplied by the time (\( \Delta t \)) the force is applied:

\[ \Delta p = F \times \Delta t \]

This derives from Newton’s second law (\( F = ma \)), expressing force as the rate of change of momentum.

**Force Definition:**
Force is a push or pull applied to an object that can change its state of motion or shape. It is a vector quantity, meaning it has both magnitude and direction.

**Characteristics of a Force:**
1. **Magnitude:** The strength or size of the force, measured in newtons (N).
2. **Direction:** The line along which the force is applied, represented by an arrow.
3. **Point of Application:** The specific point where the force is applied on an object.
4. **Contact or Action-at-a-distance:** Forces can result from direct contact between objects or act at a distance (e.g., gravitational force).

**Impulse-Momentum Formula Derivation (Short Answer):**
The impulse-momentum theorem states that the change in momentum of an object is equal to the impulse applied to it. Mathematically, this is expressed as:

Impulse(Delta p) = Force*Time

Now, considering Newton’s second law (\( F = ma \)), we can express force as the rate of change of momentum:

F = Delta p/Delta t

Rearranging, we get:

\[ \Delta p = F \times \Delta t \]

This equation is the impulse-momentum formula, where:
– \( \Delta p \) is the change in momentum,
– \( F \) is the force applied, and
– \( \Delta t \) is the time the force is applied.

What are the limitations of 2nd law of motion? Derive second law of motion.

**Limitations of Newton’s Second Law:**
1. Applicable mainly to particles.
2. Assumes constant mass.
3. Assumes force is directly proportional to acceleration.
4. Not suitable for forces depending on higher derivatives.

**Derivation of Newton’s Second Law:**
Newton’s second law (\(F = ma\)) states that force is the product of mass and acceleration. Derived from the relationship between force, mass, and acceleration, it expresses how an applied force causes a change in motion.

Limitations of Newton’s Second Law:

1. **Applicability:** Newton’s second law is strictly applicable to particles or point masses and may not be directly applicable to objects with changing shapes.
2. **Constant Mass:** It assumes constant mass, which might not be the case in certain relativistic or extreme physical conditions.
3. **Acceleration Dependency:** The equation assumes that force is directly proportional to acceleration, which might not hold in some situations.
4. **Not for All Forces:** The law doesn’t account for forces that depend on higher derivatives of position (higher than the second derivative).

**Derivation of Newton’s Second Law (Short Answer):**
Newton’s second law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration. Mathematically, it is expressed as:

\[ F = ma \]

Where:
– F is the force applied,
– m is the mass of the object, and
– a is its acceleration.

The law is derived from the fundamental principles of classical mechanics and the relationship between force, mass, and acceleration. The basic idea is that the force required to accelerate an object is directly proportional to its mass and the acceleration applied.

Differentiate between impulsive force and impulse of a force.

Characteristic	Impulsive Force	Impulse of a Force
Definition	A force applied over a very short duration, leading to a sudden change in momentum.	The product of force and the time over which the force is applied, causing a change in momentum.
Nature	Acts over an extremely brief time interval.	Represents the overall effect of a force acting over a specific time duration.
Duration	Very short duration, often considered instantaneous.	Duration is explicitly considered in the calculation (Δt in the impulse formula).
Effect on Momentum	Results in a rapid and significant change in momentum.	Represents the cumulative change in momentum caused by a force.
Mathematical Expression	F⋅ΔF⋅Δt where ΔF is the impulsive force.	F⋅ΔF⋅Δt where ΔF is the force and ΔFΔt is the time.
Example	A ball hitting a wall and rebounding in a very short time.	A constant force applied to an object over a specified time period.

An object travelling in circular motion with constant speed, determine its acceleration?

The acceleration of an object moving in circular motion with constant speed is directed toward the center of the circle and is called centripetal acceleration. Its magnitude (\(a_c\)) is given by the formula:

\[ a_c = \frac{v^2}{r} \]

where:
– \( v \) is the constant speed of the object, and
– \( r \) is the radius of the circular path.

Centripetal acceleration ensures that the object changes direction continuously while maintaining a constant speed.

What is the relation between distance and displacement?

Characteristic	Distance	Displacement
Definition	The total path length traveled by an object, always positive or zero.	The change in position of an object; it has both magnitude and direction.
Scalar or Vector	Scalar (only magnitude)	Vector (magnitude and direction)
Symbol	�d	Δ�Δr (or �s, �D)
Units	Meters (m), kilometers (km), etc.	Meters (m), kilometers (km), etc.
Path Dependency	Path-dependent; it depends on the actual route taken.	Path-independent; it only considers the initial and final positions.
Always Positive	Non-negative; distance is always equal to or greater than zero.	Can be positive, negative, or zero, depending on the direction of displacement.
Example	If you walk 5 meters to the east, then 3 meters to the west, the total distance traveled is 8 meters.	If you walk 5 meters to the east, then 3 meters to the west, the displacement might be 2 meters to the east.
Instantaneous Value	Instantaneous speed can be found by calculating the distance traveled per unit time.	Instantaneous velocity is found by calculating the displacement per unit time, considering direction.

Define radius of gyration and moment of inertia, why moment of inertia is so
important in rotational motion, what are the aspects influencing moment of inertia?
Derive moment of inertia of a circular disc

**Radius of Gyration:**
– *Definition:* \( k \) is the distance from the axis of rotation where the entire mass could be concentrated to yield the same moment of inertia.

**Moment of Inertia:**
– *Definition:* \( I \) measures an object’s resistance to changes in rotation, depending on its mass distribution around the axis.

**Importance:**
– *Significance:* Crucial in rotational motion, indicating how objects respond to external torques.

**Influencing Aspects:**
– *Factors:* Mass distribution, shape, and axis of rotation affect the moment of inertia.

**Derivation for Circular Disc:**
– *Formula:* \( I = \frac{1}{2}MR^2 \)
– *Explanation:* Reflects mass distribution in a circular disc and its resistance to rotational motion around its central axis.

Define com, torque and mention use of torque in our life. Derive the equation
of torque and show that torque depends on moment of inertia

**Center of Mass (COM):**
– *Definition:* The center of mass is the point in an object or system where its mass can be considered concentrated.

**Torque:**
– *Definition:* Torque is the rotational equivalent of force, causing an object to rotate around an axis. It depends on the force applied, the distance from the axis, and the angle.

**Use of Torque in Our Life:**
– *Example:* Turning a wrench, opening a door, or riding a bicycle involve the application of torque.

**Derivation of Torque Equation:**
– *Formula:* Torque (\(\tau\)) is given by \(\tau = r \times F \times \sin(\theta)\), where:
– \(r\) is the lever arm (distance from the axis),
– \(F\) is the force applied, and
– \(\theta\) is the angle between the force and the lever arm.

**Torque Depends on Moment of Inertia:**
– *Relationship:* Torque (\(\tau\)) is proportional to the product of moment of inertia (\(I\)) and angular acceleration (\(\alpha\)), expressed as \(\tau = I \times \alpha\).
– *Explanation:* This relationship shows that the torque required to change the rotation of an object depends on how its mass is distributed (moment of inertia).

Derive expression for KE of rotation

Derivation of Rotational Kinetic Energy (Short Answer): For an object rotating about a fixed axis, the rotational kinetic energy (rotKErot​) is given by:

rot=12   2KErot​=21​Iω2

where:

• I is the moment of inertia of the object,
• ω is the angular velocity of the object.

This equation parallels the linear kinetic energy equation (linear=12   2KElinear​=21​mv2) but involves angular quantities for rotational motion. It represents the energy associated with the rotation of an object.

11. Compare linear and angular momentum. Prove principle of conservation of
angular momentum

Characteristic	Linear Momentum	Angular Momentum
Quantity Symbol	p (linear momentum)	L (angular momentum)
Vector/Scalar	Vector	Vector
Definition	p=m⋅v (product of mass and velocity)	L=I⋅ω (product of moment of inertia and angular velocity)
Units	kg·m/s (kilogram meter per second)	kg·m²/s (kilogram meter squared per second)
Conservation Principle	Conservation of linear momentum holds in isolated systems.	Conservation of angular momentum holds in isolated systems.
Application	Applicable to objects moving in a straight line.	Applicable to objects rotating about an axis.

Mentions some applications of linear, circular and rotational motion

Applications of Linear Motion:

1. Transportation: Vehicles moving in a straight line, such as cars, trains, and airplanes, involve linear motion.
2. Projectile Motion: The motion of a thrown object, like a ball or a projectile, is a combination of linear and vertical motions.
3. Conveyor Belts: Linear motion is used in conveyor belts to transport goods.

Applications of Circular Motion:

1. Amusement Park Rides: Ferris wheels, carousels, and roller coasters involve circular motion.
2. Planetary Orbits: Celestial bodies, like planets and moons, move in circular or elliptical orbits around a central mass.
3. Spinning Objects: Objects like wheels, records, and hard drives involve circular motion.

Applications of Rotational Motion:

1. Flywheels: Used in machinery to store rotational energy and stabilize rotational speed.
2. Drilling: In drilling machines, the drill bit undergoes rotational motion.
3. Gyroscope: Used in navigation systems and devices to maintain orientation and stability.

These applications demonstrate the diverse and practical uses of linear, circular, and rotational motion in various aspects of everyday life and technology.

Linear Motion:

1. Transportation: Cars, trains, airplanes.
2. Projectile Motion: Thrown objects.
3. Conveyor Belts: Goods transportation.

Circular Motion:

1. Amusement Park Rides: Ferris wheels, carousels.
2. Planetary Orbits: Celestial bodies.
3. Spinning Objects: Wheels, records.

Rotational Motion:

1. Flywheels: Energy storage and stability.
2. Drilling Machines: Rotational motion in drilling.
3. Gyroscopes: Navigation and stability.

What are the limitations of first law of motion?

Limitations of Newton’s First Law:
1. Inertia: The first law assumes perfect inertia, but in reality, some resistance to motion exists due to factors like friction and air resistance.
2. Constant Velocity: It doesn’t explain how an object maintains constant velocity under the influence of a net force.
3. Applicability: Strictly applicable only in inertial reference frames; non-inertial frames introduce complexities.
4. Quantification: It doesn’t provide a way to quantify the force needed to overcome inertia.

These limitations highlight specific scenarios where the first law’s simplicity may not fully describe real-world dynamics.

What are the limitations of third law of motion?

Limitations of Newton’s Third Law:

1. Action and Reaction: While the law states that every action has an equal and opposite reaction, the effects of these forces might be very different due to differences in mass and acceleration.
2. Not Always Applicable: The law may not be directly applicable in situations involving variable masses, relativistic speeds, or quantum scales.
3. Friction and Air Resistance: In the presence of friction or air resistance, the equal and opposite forces might not completely cancel out.

These limitations highlight cases where the simplicity of Newton’s third law may not fully capture the complexities of certain physical interactions.

Thanks for your time!

Encoder and Decoder

 What is an encoder?
An encoder is a device that converts a physical signal into a digital signal.

The encoder is also a combinational logic circuit; it converts information, such as a
decimal number or an alphabetic character, into some coded form such as binary or BCD.

Encoders are used in a variety of applications, such as robotics, industrial control, and
telecommunications.

An encoder accepts digit on its inputs, such as a decimal or octal digit, and converts it
to a coded output, such as a binary or BCD. Encoder can also be devised to encode
various symbol and alphabetic characters. This process of converting from familiar
symbols or numbers to a coded format is called encoding

 Types of encoders: There are many different types of encoders, including:
 Optical encoders: Optical encoders use light to detect changes in position or rotation.
 Magnetic encoders: Magnetic encoders use magnetic fields to detect changes in position or rotation.
 Inductive encoders: Inductive encoders use eddy currents to detect changes in position or rotation.

 How encoders work:

Encoders typically work by detecting changes in position or rotation and converting
these changes into a digital signal.
The digital signal can then be used to control a motor, robot, or other device.

What is a decoder?
A decoder is a device, circuit, or program that converts an analogue signal into digital data.
• It is Multiple Input & Multiple Output device.
• Decoder is a combinational circuit that converts n lines of input into maximum 2^n lines
of output.
• Applications of decoders are converting binary code to other codes like:
• Binary to octal
• Binary to Hexadecimal
• Binary to decimal
 Types of decoders:
1. Audio decoders: These decoders convert compressed audio files, such as MP3s, back to
their original WAV format.
2. Image decoders: These decoders convert compressed image files, such as JPEGs and
PNGs, back to their original bitmap format.
3. Text decoders: These decoders convert encoded text, back to its original form.
4. Video decoders : These decoders convert compressed video files , such as mp4s, back
to their original Bitmap format.

5. How decoders work:
Decoders typically work by converting a digital signal into a physical signal.
The physical signal can then be used to control a motor, robot, or other device.

About Decoder Circuit
The arrangement ensures that each output line is activated when the corresponding input
combination is detected. When an input line is low (0), its complement becomes high (1),
and the corresponding AND gate outputs a low signal (0) regardless of the other input. As a
result, only one output line is activated at a time, based on the input combination.
Decoders are used for code conversions and used to convert an analogue signal into digital data, which can
be processed by a computer. Decoders are essential components in various fields including computer
science, electronics, telecommunications and information technology. Here are some of the common uses of
decoders in different contexts:

 DIGITAL ELECTRONICS:
 Memory Systems: Decoders are used to select a specific memory location RAM (Random
Access Memory) or ROM (Read-Only Memory) based on the address lines.
 Display Systems: In applications like 7-segment displays, decoders convert binary information into a format that can be displayed on the screen.
 Data Multiplexing: Decoders are used to demultiplex data from one source to multiple output lines.
 Address Decoding: Decoders help select a specific device or registers in a computer’s memory
or (Input / Output) system.
 COMMUNICATION SYSTEMS:
 Error Detection and Correction: In communication systems, decoders are used in errordetecting and error-correcting codes.
 Digital Signal Processing: Decoders are used to convert digital signals from one format to another in various signal processing applications.
 TELECOMMUNICATIONS:
 Digital Television: Decoders are used in set-top boxes to convert digital signals into analog or high
definition signals for television display.
 Mobile Communication: Decoders are used in mobile phones to decode audio and video signals for playback.
 Fax Machines and Modems: Decoders are used to interpret signals received from fax machines or modems.
 COMPUTER PROGRAMMING AND SOFTWARE:
 File Compression: Decoders are used to decompress compressed files. For instance in applications like ZIP files, decoders are employed to extract the original files.
 Encoding and Decoding Data: Decoders play a crucial role in encryption and decryption
algorithms, ensuring secure communication over the internet.
 Media Codecs: Decoders are used in media players to decode audio and video files into formats that
can be played back and displayed on a computer or mobile device.
 AUTOMOTIVE INDUSTRY:
 In-Car Entertainment Systems: Decoders are used in multimedia systems within cars to decode
audio and video signals for entertainment purposes.
 Engine Control Units: Decoders are utilized in the electronic control units of modern vehicles to
interpret sensor data and make necessary adjustments to engine performance.
 ROBOTICS AND AUTOMATION:
 Sensor Data Interpretation: Decoders help interpret signals from sensors, enabling robots to react
their environment effectively.
 Control Systems: Decoders are used in control systems to interpret digital signals and control
various actuators and devices.

JavaScript Learning Tutorial P001

Introduction

JavaScript is one of the most essential programming languages in web development. Whether you’re looking to build dynamic websites, interactive web applications, or even mobile apps, mastering JavaScript is a must. In this guide, we will take you through the fundamentals of JavaScript and provide the best resources to help you become proficient in no time.

1. Understanding JavaScript: What Is It?

JavaScript (JS) is a lightweight, interpreted programming language that allows developers to add interactivity and functionality to web pages. It is widely used alongside HTML and CSS to create engaging user experiences.

Key Features of JavaScript:

• Client-side and server-side capabilities
• Versatile and beginner-friendly
• Supports object-oriented, procedural, and functional programming
• Large ecosystem and community support

---

2. Setting Up Your JavaScript Environment

Before writing your first JavaScript code, you must set up a development environment.

• Web Browser: Modern browsers like Chrome, Firefox, and Edge come with built-in JavaScript engines.
• Code Editor: Visual Studio Code, Sublime Text, or Atom are excellent choices.
• Console & DevTools: Use browser developer tools (F12 or right-click -> Inspect) to test and debug JavaScript code.

---

3. JavaScript Basics: Getting Started Let’s go over some basic concepts that every beginner should know:

• Variables & Data Types:let name = "Jakaria"; // String let age = 25; // Number let isStudent = true; // Boolean
• Functions:function greet(name) { return "Hello, " + name + "!"; } console.log(greet("Alice"));
• Loops & Conditional Statements:for (let i = 1; i <= 5; i++) { console.log(i); }

---

4. Essential JavaScript Concepts to Master As you advance, focus on these important JavaScript concepts:

• DOM Manipulation: Interact with HTML elements dynamically.
• Events & Event Listeners: Handle user interactions.
• ES6+ Features: Learn modern JavaScript features like arrow functions, template literals, and destructuring.
• Asynchronous JavaScript: Understand promises, async/await, and callbacks.
• APIs & Fetch: Communicate with web servers using REST APIs.

---

5. Best Free Resources to Learn JavaScript Here are some of the best places to learn JavaScript for free:

• MDN Web Docs
• JavaScript.info
• freeCodeCamp
• Eloquent JavaScript (Book)

---

JavaScript is a programming language used to create dynamic content for websites.

It is a lightweight, cross-platform, and single-threaded programming language.

JavaScript is an interpreted language that executes code line by line providing more flexibility.

JavaScript is the world’s most popular programming language.

Why Study JavaScript?

JavaScript is one of the 3 languages all web developers must learn:

1. HTML to define the content of web pages

2. CSS to specify the layout of web pages

3. JavaScript to program the behavior of web pages

HTML adds Structure to a web page, CSS styles it and JavaScript brings it to life by allowing users to interact with elements on the page, such as actions on clicking buttons, filling out forms, and showing animations.

---

<!DOCTYPE html>
<html>
<body>

<h2>JavaScript in Body</h2>

<p id=”demo”></p>

<script>
document.getElementById(“demo”).innerHTML = “My First JavaScript”;
</script>

</body>
</html>

---

Conclusion JavaScript is an exciting and powerful language that opens up endless possibilities in web development. By practicing consistently and using the right learning resources, you can become a proficient JavaScript developer. Keep experimenting, build projects, and never stop learning!

Are you currently learning JavaScript? Share your experiences in the comments below!

Python Basic Concept

Top 10 Online Python Compiler Picks 2024

• Coding Ninjas
• PyDev
• PyCharm
• Jupyter Notebook
• Atom
• Spyder
• IDLE
• Sublime Text
• Vim
• Visual Studio Code

Matrix Math Basic Concept

Matrix:
A matrix is a rectangular arrangement of numbers, expressions, or symbols in rows and columns enclosed by ( ), [ ],  or || ||.

Row: The row elements are horizontally arranged

Column: The column elements are vertically arranged.

What is the Dimension of a Matrix?
The dimensions, or size, of a matrix, are defined as, [Number of Rows * Number of Columns ]

A matrix is usually denoted by a capital letter and the elements within the matrix are denoted by lowercase letters,

Row Matrix:

A matrix is said to be a row matrix if it has only one row. A row matrix has only one row but any number of columns.

Column Matrix:

A matrix is said to be a column matrix if it has only one column. A column matrix has only one column but any number of rows.

Square Matrix:

A square matrix has the number of columns equal to the number of rows.

Rectangular Matrix:

A matrix where number of rows is not equal to the number of columns is called rectangular matrix.

Diagonal Matrix:

A square matrix is said to be a diagonal matrix when – 1. aij != 0 (i=j) | 2.  aij = 0 (i!=j)

A square matrix where all the elements are zero except those on the main diagonal.

Scalar Matrix:

A square matrix is said to be a scalar matrix when – 1. aij = k (i=j) | 2.  aij = 0 (i!=j)

A diagonal matrix is said to be a scalar matrix if all the elements in its principal diagonal are equal to some non-zero constant.

Identity/Unit Matrix:

A square matrix is said to be a scalar matrix when – 1. aij = 1 (i=j) | 2.  aij = 0 (i!=j)

An identity matrix is a square matrix that has 1’s along the main diagonal and 0’s everywhere else and denoted by I.

Triangular Matrix:

A square matrix is said to be a triangular matrix when – 1. aij = 0 (i<j) | 2.  aij = 0 (i>j)

A square matrix whose elements above or below the main diagonal are all zero

Upper Triangular Matrix:

A square matrix is said to be a upper triangular matrix when – 1. aij = 0 (i>j)

A square matrix whose elements below the main diagonal are all zero.

Lower Triangular Matrix:

A square matrix is said to be a upper triangular matrix when – 1. aij = 0 (i<j)

A square matrix whose elements above the main diagonal are all zero.

PERMUTATION MATRIX
A square matrix is said to be a permutation matrix when –
1. Have to be a square matrix
2. Each entry either 1 or 0
3. Each row contain a significance one  (1)
4. Each column contain a significance one  (1)

Sparse Matrix:

A square matrix is said to be sparse matrix when – 1. Non-zero > Zero

Dense matrix:

A square matrix is said to be dense matrix when – 1. Zero > Non-zero

Scalar Multiplication: It is defined with, “multiply each element in the matrix by the given scalar”

Matrix Addition & Subtraction: Two matrices can only be added or subtracted if and only if they have the same dimensions

Matrix Multiplication: Condition of matrix multiplication is, “Matrix multiplication between two matrices will be possible if and only if the number of columns of the first matrix is equal to the number of rows of the second matrix

Idempotent Matrix: The matrix A is idempotent matrix if and only if A.A=A

Involuntary matrix: If A.A = I or A^-1 = A then the matrix A is said to be an involuntary matrix.

Nilpotent matrix: A nilpotent matrix is a square matrix N such that A.A=0 for some positive integer k. The smallest such k is called the index of N, sometimes the degree of N.

Thank You!

Picture of Math Number System

Classification and Definition of Number

@. Complex Number:

Complex number is a number that comprises both a real part and an imaginary part.

#. It is expressed in the form: a + bi

a	a represent the real part.
b	b represent the imaginary part.
i	i is the imaginary unit.

@. Complex Number:  C

An imaginary number is a mathematical concept that extends the set of real number.

Such as:{  }

@. Real Number: R

Real numbers encompass both rational and irrational numbers.

They represent the complete continuum of numbers on the number line, including all possible function, decimal, and integers.

Such as:

@. Rational Number: Z

Rational numbers include integer number and fraction number.

Such as:               {-2, -1, 0, 1, 2} (Integer Number)

@. Irrational Number:

Irrational Numbers are real numbers that cannot be expressed as fraction.

Such as:

@. Integer Number:

Integer numbers are the whole numbers along with their negative counterparts.

Such as:{-2, -1, 0, 1, 2}

@. Fraction Number:

A fraction is a number that represents a part of a whole.

It is written in the form of two numbers separated by a line, where the number above the line is called the numerator, and the number below the line is called the denominator.

Such as:

@. Natural Number: N

Natural numbers also known as counting number, are set of positive integers.

Such as:{1, 2, 3, 4, 5, 6, 7, 8}

@. Whole Number:

Whole numbers include all the natural number with zero.

Such as:{0, 1, 2, 3, 4, 5, 6, 7, 8}

Java Variables and Modifiers | Java Programming

Variables in java

Local and Instance Variables
Class or Static Variables

1. Local variables

• Local variables are declared inside methods, constructors, or blocks.
• Local variables are visible only within the declared method, constructor or block.
• Access modifiers cannot be used for local variables.

2. Instance variables

• Instance variables are declared in a class, but outside a method, constructor or any block.
• Instance variables are created when an object is created with the use of the keyword ‘new’ and destroyed when the object is destroyed.
• Access modifiers can be given for instance variables.
• The instance variables are visible for all methods, constructors and block in the class.

3. Class/static variables

• Class variables also known as static variables are declared with the static keyword in a class,
but outside a method, constructor or a block.
• Static variables are rarely used other than being declared as constants.
• Visibility is similar to instance variables.
• However, most static variables are declared public since they must be available for users of the class.

❑ Local Variable
A variable that is declared inside the method
❑ Instance Variable
A variable that is declared inside the class but outside the method.
It is not declared as static.
❑ Static variable
Similar to instance variable but it is declared as static.

Example to understand the types of variables

---

class A
{
int data=50; // instance variable
static int m=100; // static variable
void method()
{
int n=90; // local variable
}
}

---

Example – 1:

---

public class MyClass
{
int a = 50;
static int b = 100;
public void method1()
{
int n = 90;
System.out.println(n);
System.out.println(a);
System.out.println(b);
}
public void method2()
{
System.out.println(n);
System.out.println(a);
System.out.println(b);
}

---

Example – 2:

---

public static void main(String[] args)
{
MyClass ob = new MyClass ();
ob.method1 ();
ob. method2 ();
}
}

---

Example – 3:

---

public class NewClass {

int max = 100; //instance variable
static int var = 50; // static variable

public static void main(String[] args)
{
int a = 10, b = 20; // local variable
System.out.println(a+b);
NewClass obj = new NewClass();
System.out.println(obj.max);
System.out.println(var);
sum();
}
public static void sum()
{
NewClass obj = new NewClass();
System.out.println(obj.max);
System.out.println(var);

//System.out.println(a+b);
}
}

---

Modifiers in java

Modifiers are keywords that you add to those definitions to change their meanings.

There are two types of modifiers in java:
1. Access modifiers
2. Non-access modifiers

1. Access Modifiers in java

The access modifiers in java specifies accessibility (scope) of a data member, method, constructor or class.

There are 4 types of java access modifiers:

1. private
2. default
3. protected
4. public

1) private access modifier

The private access modifier is accessible only within the class.
Simple example of private access modifier:
In this example, we have created two classes A and Simple. A class contains private data member and private method. We are accessing these private members from outside the class, so there is compile time error.

2) default access modifier

• If you don’t use any modifier, it is treated as default by default.
• The default modifier is accessible only within package.

Example of default access modifier:
In this example, we have created two packages pack and mypack. We are accessing the A class from outside its package, since A class is not public, so it cannot be accessed from outside the package.
In the above example, the scope of class A and its method msg() is default so it cannot be accessed from outside the package.

3) protected access modifier

• The protected access modifier is accessible within package and outside the package but through inheritance only.
• The protected access modifier can be applied on the data member, method and constructor. It can’t be applied on the class.

Example of protected access modifier:
In this example, we have created the two packages pack and mypack. The A class of pack package is public, so can be accessed from outside the package. But msg method of this package is declared as protected, so it can be accessed from outside the class only through inheritance.

4) public access modifier

• The public access modifier is accessible everywhere.
• It has the widest scope among all other modifiers.

2. Non-Access Modifiers

static modifier
for creating class methods and variables
final modifier
for finalizing the implementations of classes, methods, and variables.
abstract modifiers
for creating abstract classes and methods.
synchronized and volatile modifiers
which are used for threads.

Thank you!

C++ Programming Basic Concept

Basic of C++ Programming

@. Father: Bjarne Stroustrup.

@. Institution: Bell Laboratory.

@. From: C -> C++

@. Use: Making operating system and gaming.

@. Some Topics of C++ language:

1. Nested loop                         06. Structure
2. If (condition)                      07. Enam
3. Function                              08. Pointer
4. String                                   09. Recursion
5. Array                                    10. Header file

@. We cannot use 32 word

auto, break, case, char, const, continue, default, do, double, else, enum, extern, float, for, goto, if, int, long, register, return, short, signed, sizeof, static, struct, switch, typedef, union, unsigned, void, volatile, while.

@. For Comment in code

Single line comment : \\ Md Jakaria Nur

Multiple line comment : \* I am a software designer and developer… *\

Skeleton of C++ Programming

#include <iostream>
using namespace std;
int main () {
cout << “ Hello Jakaria!\n”;
return 0; }

01. To print backspace = \b
02. To print form feed = \f
03. To print new line = \n
04. To print carriage return = \r
05. To print horizontal tab = \t
06. To print double quote = \"
07. To print single quote = \'
08. To print null = \0
09. To print backslash = \\
10. To print vertical tab = \v
11. To print alert = \a
12. To print question mark = \?
13. To print octal constant = \N
14. To print hexadecimal constant = \xN

Data Type and Variables

No	Data Type	Keyword	Size in bytes	Range	Place hold
01	Character	char	1	-128 to 127	%c
02	Unsigned Character	Unsigned char	1	0 to 255	%c
03	Integer	int	2 or 4	-32768 to 32767 or-2147483648 to 2147483647	%d
04	Unsigned Integer	Unsigned int	2 or 4	0 to 65535 or0 to 4,294,967,295	%u
05	Long Integer	long int	4	-2147483648 to 2147483647	%d
06	Long Long Integer	long long int	8	(-2^63) to (-2^63-1)	%d
07	Float	float	4	1.2E-38 to 3.4E +38	%f
08	Double	double	8	2.3E-308 to 1.7E +308	%f

Use some variables

#include <iostream>
using namespace std;
int main () {
int a, b, sum, sub;
cout << "Enter your first number: ";
cin >> a;
cout << "Enter your second number: ";
cin >> b;
sum = a+b;
sub = a-b;
cout << "Summation = " << sum << endl;
cout << "Subtraction = " << sub << endl;
return 0; }

Details on Data Type

@.     Character – %c

Character is worked by only alphabet. Its value is in 0 – 256.

Only 25 letters write by Character.

Signed char and Unsigned char = 1 byte

@.     Integer – %d

Integer is worked by integer numbers. Its value is in 0 to 65535.

int and Unsigned int = 2 bytes

@.     Float – %f

Integer is worked by Fraction and decimal numbers.

It is showed only 6 digits.

Float = 4 or 32 bytes

@.     Double: %lf

Double is worked by big decimal and scientific numbers more than float numbers.    (6.44, 8.11, 2*10^2, 5.2*10^14)

It is showed only 16 digits.

To input data (use of scanf)

#include <stdio.h>
int main () {
int a;
int b;
Printf("Enter First Integer Number: ");
scanf("%d", &a);
Printf("Enter Second Integer Number: ");
scanf("%d", &b);
printf("Sum Two Integrer = %d", a+b);
return 0; }

Condition (use of if and else)

#include <stdio.h>
int main () {
int x;
Printf("Enter  Your Number: ");
scanf("%d", &x);
if (x>60); {
printf("Md Jakaria Nur\n");
}else{
printf("Md Jubayer Nur\n");}
return 0; }

Condition (use of if and else if)

#include <stdio.h>
int main () {
int age;
printf("Enter value of age: ");
scanf("%d", &age);
if (age<2);{
printf("Intant\n");}
else if (age<10);{
printf("Child\n");
} else if (age<20);{
printf("Teenage\n");
} else if (age<30);{
printf("Adult\n");
} else {
printf("%d\n"); }
return 0; }

Switch Case

#include <stdio.h>
int main () {
int x;
printf("Emter One Integer Number: ");
scanf("%d,&x");
switch(x){
case 1:
printf("Value is 1\n");
printf("Welcome!\n");
break;
case 2:
printf("Value is 2\n");
break;
case 3:
printf("Value is 3\n");
break;
default:
printf("Unknown!\n"); }
return 0; }

@. Case 1 = case 2 +4 use possible but case 2 + x is not possible.

@. Switch case just char and int use possible  but float and double use is not possible.

Operators, Precedence and Associativity

x = y + z
y = 5 - 5
Here,
Operator:  +, -, =, *, /, %, ++, -
Variable: x, y, z

Arithmetic Operators:

These are the operators used to perform arithmetic or mathematical operations on operands.

#. Arithmetic Operators are two types:

1. Unary Operator
2. Binary Operator

#. Unary Operator:

Operators that operates or works with a single operand are unary operators.

Such as: (z++, -x)

#. Binary Operator:

Operators that operates or works with two operand are binary operators.

Such as: (+, -, *, /)

#. Relation Operator

Relation Operators are used for compassion.

Checking if one operand is equal to other operand or not, an opearand is gratien than other operand or not etc.

Such as: (==, >=, <=)

@. Logical Operators

Logical Operators are used to combine two or more conditions or constraints or to complement the evaluation of the original condition in consideration.

The result of a logical operator is a boolean value either true of false.

Thank You!