# 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 t_{1}, t_{2}, t_{3} secs, are respectively v_{1}, v_{2}, v_{3} then we find,

v_{1}/t_{1} = v_{2}/t_{2} = v_{3}/t_{3}

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 ∞ t ^{2}.**

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 2^{2} x h in 2 sec., 3^{2} 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 h_{1}, h_{2,} and h respectively, in t_{1}, t_{2}, t_{3} secs., then,

h_{1}/t_{1}^{2} = h_{2}/t_{2}^{2} = h_{3}/t_{3}^{2}

so, h ∞ t^{2}.

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)

Speed =

• SI Unit- ms-1

Speed

VELOCITY

• is the rate of

change of

displacement

• Is a measured

speed in a given

direction

• It tells us not only

the speed of the

object but also the

direction

• It is a vector

quantity.

• Velocity =

Speed & Velocity

Speed & Velocity

Example A particle moves along the x-axis in such a way that its coordinates x varies with time ‘t’

according to the equation x = 2 – 5t + 6t2. What is the initial velocity of the particle?

Solution

The negative sign implies that at t = 0 the velocity of the particle is along negative x direction.

Solved Example

A swimmer’s speed in the direction of flow of a river is 12 km h-1. Against the direction of flow of the river

the swimmer’s speed is 6 km h-1. Calculate the swimmer’s speed in still water and the velocity of the river

flow.

Solution

Let vs and vr , represent the velocities of the swimmer and river respectively with respect to ground.

When the river flow and swimmer move in the same direction, the net velocity of swimmer is 12 km p h-1.

Relative Velocity in One and Two Dimensional

Motion

Comparison Basis Velocity Acceleration

Meaning It alludes to the speed of an object in

the given direction.

Acceleration implies to any change in the

velocity of object with respect to time

Calculated With Displacement Velocity

What is it’s

Nature?

Vector Vector

What is it? Rate of change of displacement Rate of change of Velocity

Formula Displacement/Time (d/t) Velocity/Time (v/t)

Ascertains How fast an object is moving and in

which direction

How fast an object’s velocity changes with

time.

Unit of

Measurement

meter/second (m/s) meter/second2

(m/s2

)

When an object is projected vertically up from the earth, we regard the upward direction as the positive direction. The

force of gravity produces acceleration downwards on the object. Acceleration is therefore taken as negative.

Equations of motion: Let u = initial velocity, v = final velocity, a = acceleration, t = time and s = displacement.

Equations of Motion

Motion and Acceleration

Motion Type Equations Characteristics

Non-accelerated Motion

a=0 x = x₀ + v * t Constant velocity (v)

Constant speed (magnitude of velocity)

No acceleration (a = 0)

Straight-line motion

Velocity (v) remains unchanged with time (constant)

Uniformly Accelerated

Motion x = x₀ + v₀ * t + (1/2) * a * t² Changing velocity (v)

v = v₀ + a * t Changing speed (magnitude of velocity)

Constant acceleration (a = constant value)

Motion may be curved or straight

Non-uniformly Accelerated

Motion x = x₀ + v₀ * t + (1/2) * a(t) * t² Changing velocity (v)

v = v₀ + ∫(a(t) dt) Changing speed (magnitude of velocity)

Acceleration (a) varies with time (non-constant)

Motion may be curved or straight

Problem: What is the displacement from A to B?

DYNAMICS = f(Kinematics+Force)

Newton’s Laws of Motion

• Force is a kind of impact, external or tends to change the inertia of any object. It

is a push or a pull, or any action that has the ability to change an object’s

situation.

• 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. The unit of force is

Newton (N).

• FOUR BASIC FORCES:

1. The gravitational force, which originates with the of matter.

2. The electromagnetic force, which includes basic electric and magnetic

interactions and is responsible for the binding of the atoms and structure of

solids

3. The weak nuclear force, which causes certain radioactive decay processes and

certain reactions among the fundamental particles,

4. The strong force, which operates among the fundamental particles and is

responsible for binding the nucleus together

FORCE

Balanced and Unbalanced force

Balanced force Unbalanced force

1. When two or more forces act

on a body and produce a net force

equal to zero then the forces are

called balanced forces.

1. When two or more forces act

on a body and produce a net force

not equal to zero then the forces

are called unbalanced forces.

2. A balanced force does not

produce any change in the state

of uniform motion or the rest of

the body.

2. An unbalanced force can

produce the change in the state of

uniform motion or the rest of the

body.

3. A balanced force does not

cause to accelerate a body.

3. An unbalanced force can

accelerate a body.

• An object at rest will stay at rest, and an object

in motion will stay in motion at constant

velocity, unless acted upon by an unbalanced

force.

• 1st part —- Inertia

• 2nd part — force

INERTIA is the tendency of an object to resist changes in its velocity: whether in motion or motionless.

• Once airborne, unless acted on by an unbalanced force

(gravity and air – fluid friction), it would never stop!

Newton’s 1st law: Law of Inertia

• 2nd Law – Newton’s second law the rate of change of momentum of

a body is proportional to the impressed force and it takes place in

the direction of the straight line in which the force acts

• Force = mass * acceleration.

• Momentum is so important for understanding motion that it was

called the QUANTITY OF MOTION by physicists such as Newton.

Force influences momentum

• Momentum can be thought of as the “power” when an object is

moving, meaning how much force it can have on another object.

• The quantity of motion mean, how dangerous could an object of

mass moving with velocity be. When there is change in the

momentum of a given body, there is change in its velocity, i.e. there

is acceleration.

Newton’s second law: Law of Acceleration/Momentum

First Law (Law of Inertia):

First Law is a special case of 2nd law of motion:

Derivation of Newton’s second law

Derivation of the first law

from the second law

• FOR EVERY ACTION THERE IS AN EQUAL AND OPPOSITE

REACTION.

• that is for every acting force there is an equal but opposite reacting

force

If the duration of the action or reaction force is t then

F x t = -Rxt i.e. IMPULSE OF ACTION = – IMPULSE OF REACTION

• this is applicable for objects either at rest or in motion

• EXAMPLES OF THIRD LAW firing of a bullet from a gun jumping from

a boat working

Third Law (Action-Reaction): Impulse

Impulsive force and impulse of a force

•IMPULSIVE FORCE (F) is a force of very high

magnitude which acts for a very short time.

•IMPULSE (J): the product of the impulsive force and

that time during which the force acts is called

impulse. It is denoted by J which is a vector quantity

J= Ft

=m a t

Explanation: Impulsive force and impulse of a force

Relation Between Impulse And Momentum

we know

Impulse-momentum formula

Impulse-momentum formula is obtained from impulsemomentum theorem which states that change in momentum of

an object is equal to impulse applied on the object. The

formula is given as follows:

Impulse-momentum formula J=Δp

When the mass is constant FΔt=mΔv

When the mass is varying Fdt=mdv+vdm

As the SI unit of impulse and momentum are equal, it is given

as Ns=kg.m.s

-1

A force of 100N acts on a stationary object of mass 10kg. After 5 sec the force

does not act, find out the distance the object travels in 10sec. From

beginning.

A body of mass 0.05kg hits a vertical wall with a horizontal velocity of 0.2m/s

and rebounds with a velocity of 0.1m/s, find the impulse

Problem impulse

Problem : A batsman knocks back a ball straight in the direction towards

the bowler without altering its initial speed of 12 m/s. If the mass of the

ball is 0.15kg, calculate the impulse imparted to the ball?

Answer: Known: vi

(Initial Velocity) = 12 m/s,

vf

(Final Velocity) = -12m/s, m (mass) = 0.15kg,

J (Impulse) = ?

Impulse is articulated as

J = mvf – mvi

= m(vf – vi

)

= 0.15 Kg (-12 -12) m/s

= -3.6N.

Collisions and Principal of conservation of momentum

if no external forces act on a system of quality objects that total momentum of the

objects in a given direction before collision is equal to total momentum in the same

direction after collision

Differences Between Elastic And Inelastic Collision

ELASTIC COLLISION INELASTIC COLLISION

1. A state where there is no net loss in kinetic

energy in the system as the result of the

collision is called an elastic collision.

2. The total kinetic energy is conserved.

3. Momentum does not change.

4. No conversion of energy takes place.

5. Highly unlikely in the real world as there is

almost always a change in energy. An

example of this can be swinging balls or a

spacecraft flying near a planet but not

getting affected by its gravity in the end.

1. A type of collision where this is a loss of kinetic energy

is called an inelastic collision. The lost kinetic energy is

transformed into thermal energy, sound energy, and

material deformation.

2. The total kinetic energy of the bodies at the beginning

and the end of the collision is different.

3. Momentum does not change.

4. Kinetic energy is changed into other energy such as

sound or heat energy.

5. This is the normal form of collision in the real world.

6. An example of an inelastic collision can be the

collision of two cars.

A bowling ball of 35.2kg, generates 218 kg.m/s units of

momentum. What is the velocity of the bowling ball?

V = p/m = 218/35.2 = 6.2m/s

10 Newton force acts on a body at rest of mass 2 kg. If after

for second the force does not act on the body then how far

will the body move from the start in 8 second

A body was at rest. A force of magnitude 15 Newton acted on it for 4

seconds and then stopped. The body afterwards travel a distance of 54 m in

9 seconds. Find the mass of the body.

A bullet of mass 10 gram is fired from a gun of mass 5 kg. If the gun

recoils at the speed of 60 cm per second calculate the initial speed of

the bullet. The bullet stops after penetrating 50 cm into a target. Find

the opposite force to the motion of the bullet.

A bullet of mass 0.01 kg comes out at a speed of 300

m per second from a gun of mass 6 kg. Find the

backward velocity of the gun.

Two bodies of masses for 40kg and 60 kg are moving opposite to each

other with velocity’s 10 m per second and 5 m per second

respectively and at a time they collide. Combined to form a single

body. With what velocity the combined body will move

A bullet of mass 10 g is fired from a gun of mass 5 kg. If the gun requires at

the speed of 60 cm per second calculate the initial speed of the bullet. The

bullet stops after penetrating 50 cm into a target. Find the opposite force to

the motion of the bullet.

Newton’s 3rd law: Different Types of Actions and Reactions

Different types of forces are generated due to action and reaction. According to the nature

different names have been assigned to this forces.

1. PULL when and object is bolt by applying a force along length then the force is called pull .

2. TENSION: In physics, a tension force is a force that develops in a rope, thread, or cable as

it is stretched under an applied force. The force which is created due to the attachment of

one object to the other is called tension. If and iron ball is present at the end of a

thread the ball pools the thread downward. This is action, it is called tension. According to

Newton’s third law the thread holes the ball upward with equal force. This is reaction. The

magnitude of the reaction force generated in the thread is equal to the weight of the ball.

This generated force is called tension.

3. PUSH first applied in front of an object is called push in opening a door from outside force

that we apply is called push.

4. ATTRACTION OR REPULSION these 2 forces act at a distant. Similar charges or to similar

magnetic poles repel each other and to de similar charges or poles attract it other without

any physical contact between them.

5. FRICTION when an object moves are tense to move over another object a frictional force is

generated between the contact service hostel this force is called friction.

• Mechanics provides

fundamental principles to

understand and predict the

behavior of physical

systems.

• Its applications are vast and

diverse, making it a

cornerstone of modern

science and engineering

• First law gives the definition of force

• second law gives a measure of force

• Third law specifiers the property of force

**01. 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.

**02. 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.

**03. 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.

04. 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.

**05. 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 |
$F⋅Δt$ where Δ$F$ is the impulsive force. | $F⋅Δt$ where Δ$F$ is the force and $Δt$ 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. |

06. 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 |
$d$ | $Δr$ (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. |

08. 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.

09. 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).

11. 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 ($KE_{rot}$) is given by:

$KE_{rot}=21 Iω_{2}$

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 ($KE_{linear}=21 mv_{2}$) 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. |

**12. Mentions some applications of linear, circular and rotational motion**

**Applications of Linear Motion:**

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

**Applications of Circular Motion:**

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

**Applications of Rotational Motion:**

**Flywheels:**Used in machinery to store rotational energy and stabilize rotational speed.**Drilling:**In drilling machines, the drill bit undergoes rotational motion.**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.

21. 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.

**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.

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