# Laws of motion - CBSE CLASS 11

Laws of motion-CBSE CLASS 11

CONTENT LIST

## Laws of Motion

Motion is one of the significant term in physics. If an object changes its position with time than it is said to be in motion. Everything in the universe moves. It might be very fast or very very slow.

Newton's laws of motion are three basic laws which describe the relationship between the motion of an object and the forces acting on it.

## Newton’s First Law of Motion

Newton's first law says that, an object is at rest tends to stay at rest, and an object is in motion tends to stay in motion, with the same direction and speed until and unless an external unbalance force is applied on it.

To move a body from rest or to make a moving body to rest, we require an unblance external force on it.

If an object is going in a specific direction, unless something happens to the object, then the object will always go in that direction. Forever.

Example: If an astronaut throws the object in space, then the object will go in motion forever, until and unless an external force is applied on it.

### Inertia and mass

Let us consider two objects of unequal masses, say a toy car and a cycle. To achive same speed on both the object we have to apply more force on the heavier object (cycle) then lighter object (toy car).

Here we notice that cycle has resisted more than toy car, or cycle has more inertia than toy car.

Thus we can say, mass is mersaurement of inertia.

### Inertia and its Types

There are three types of inertia:

(a) Inertia of rest

(b) Inertia of motion

(c) Inertia of direction

### Inertia of rest

If an object is at rest tends to stay at rest, this property of object is said to be inertia of rest.

Example: If a person is sitting in a car fall backward when it starts suddenly due to inertia of rest.

### Inertia of motion

If an object is in motion tends to stay in motion, this property of object is said to be inertia of motion.

Example: A man jumping from moving train falls forward due to inertia of motion.

### Inertia of direction

If an object is in motion in a particular direction it tends to remain in same direction, this property of object is said to be inertia of direction.

Example: When a car makes a sharp turn at a high speed, the driver tends to get thrown to other side due to inertia of direction.

## Frame of Reference

Consider a situation that a car is moving on the road with some velocity 40km/hr east. If We ask , what is the speed of the car? A person standing on the ground would say it’s moving with some velocity 40km/hr east. On the other hand a person seating in the car will say it is at rest.If there is another person ,Who is riding on a bike with velocity 30 km/hr ,Will give answer as 10 Km/hr. Here we observe that the answer depends on the observer. So to study the motion of any body , first we have to define the observer and frame of reference.

To describe the motion of any object we need a reference system with origin and cooridinate axes, it is called frame of reference.

Frame of referenace are of two types:
(i) Inertial frame of reference
(ii) Non-inertial frame of reference

### Inertial frame of reference

A frame of reference which is at rest or which is moving with a uniform velocity along a straight line is known as an inertial frame of reference. In an inertial frame of reference the law of inertia and other laws of physics are valid.

Inertial frame of refrence has no acceleration, i.e., a = 0.

In ideal situation, there is no inertial frame of reference in the universe. But we consider earth as an inertial frame of reference for all kind of observations except observation of motion of planets. In order to observe motion of planets,the sun can be assumed as an inertial frame of reference.

### Non-inertial frame of reference

A non-inertial frame of reference is a frame of reference in which the law of inertia does not valid. It undergoes acceleration with respect to an inertial frame.

Example: A car at rest on a traffic light. When the traffic light turns green and the car accelerates forward. During this acceleration, the car is in a non-inertial frame of reference.

## Momentum

Momentum is a vector physical quantity, it is measured by the product of mass and velocity of the object. It is denoted by P.

P = mv (Where m = mass of the object, v = velocity of the object)

The direction of momentum is same the direction of velocity of the object.

The SI unit of the momentum is kgm/s and the dimension of the momentum is [MLT^-1].

## Force

A force is a physical entity which changes or try to change the state of inertia of an object. It is determine by the product of mass and acceleration of the object. It is denoted by F.

F = ma (Where m = mass of the object, a = acceleration of the object)

The SI unit of the force is N or kgm/s² and the dimension of the force is [MLT^-2].

### Balance Force

When two or more forces are acting on the object, if sum of all x, y, z components is equal to 0 then the forces acting on the object are said to be balanced force.

### Unbalanced Force

When two or more forces are acting on the object, if sum of all x, y, z components is not equal to 0 then the forces acting on the object are said to be unbalanced force.

### Resultant Force

When two or more forces are acting on the object, then the single force which produces the same effect as produced by all the forces acting together is called resultant force.

## Newton's Second Law of Motion

According to Newton's second law , The acceleration of an object depends upon two variables

(i) The net force acting on the object
(ii) The mass of the object

The acceleration of an object is directly proportional to the net force acting on the object and inversely proportional to the mass of the object.

### Mathematical Formulation of Second Law

Consider an object of mass = m , net force acting on it = F and acceleration produced = a.

Acccording to Newton's Second Law of Motion,

a∝F -----eqn-1

a∝1/m -----eqn-2

Combing eqn-1 and eqn-2 We get,

a∝F/m

a=K.F/m Where k is propotionality constant and it's value is 1.

a=F/m or F=ma

### Application of Newton's Second Law of Motion

Consider an object of mass m is moving with velocity u. When a force F applied on it than it velocity becomes v after time t.

Initail momentum = mu

Final momentum = mv

Change in momentum= mv-mu

Rate of change of momentum = (mv-mu)/t

Rate of change of momentum = m(v-u)/t

Rate of change of momentum = ma (a = acceleration in object due to force F)

Rate of change of momentum = F

The net Force applied on the object is equal to the Rate of change of momentum .

## Impulse

The product of the magnitude of a force applied on a object and the time for which it is applied, is called impulse of the force.

It is represented by J, and the unit is kgm/s or Ns.

## Newton's third Law of Motion

Newton's third law states that, every action has its equal and opposite reaction. These action and reaction forces act on two different bodies.

If an object A exerts a force F on another body B, then B exerts a force (-F) on A, these forces acting along the line joining the bodies.

Example: While walking on the ground, we push the ground backward (action) by our feet. The ground also exerts a forward force(reaction) on our feet of equal magnitude in the opposite direction which makes us move forward.

### Action and Reaction

Action and Reaction Forces

Action and Reaction forces are exerted by different objects.

Action and Reaction forces always occur simultaneously means they are always in pair.

### Applications of Third Law

Firing a Bullet from a Gun

When a bullet is fired from a gun, the gun exerts a force (action) on the bullet in the forward direction. The bullet also exerts an equal force (reaction) on the gun in the backward direction.

Motion of a Boat in Water

When the boatman pushes the water backward by his oar (action ) the water pushes the boat forward (reaction).

Rockets and Thrust

Rocket engine produces hot exhaust gases that flow out of the back of the engine (action). A thrusting force is produced in the opposite direction is reaction.

## Significance of Newton’s Laws

Newton's law of motion is very important to understand the basic laws of nature.

Law of inertia states that an object gains or stops its acceleration when a force is acting on it.Any motion o object is due to net force acting on it.

Second law of motion helps us to determine the maganitude of force that is equal to product of mass of object and acceleration developed it due to force.This concept help us to define weight. Weight is the force by which the Earth pulls an object.

Third law of motion tells us that every action has an equal and opposite reaction. With this concept we understant how we walk and swim. How rocket propells etc.

Also law of momentum and gravitation are too important for us to understand the nature of Earth.

## What is Pulley?

A pulley is a mechanical device consist of wheel on an axle and a string/rope is moved along periphery of pulley in a grooved wheel to facilitate the movement of the objects attached to it. We can easily lift heavy objects with the help of a pulley.

How Does a Pulley Work?

Pulley changes the direction of the force, making it easier to lift something by applying force. The wheel rotates about its axle when the rope is pulled from one end and the object rises up which is attached at other end of rope. The groove in the wheel keeps the rope in place.

Mechanical Advantage of a Pulley: It measures the efficiency of the pulley and equal to the ratio of load lifted and applied force (effort).

There are three types of pulley.

1. Fixed Pulley
2. Movable Pulley
3. Compound Pulley

### Fixed Pulley

Fixed Pulleys are attached to a rigid/fix structure and they are not movable.

### Movable Pulley

A movable Pulley is not attached to a fixed/rigid structure. One of the ends of the rope is connected to a rigid structure only another end passes through pulley. In movable Pulley, wheels carry the load instead of ropes. If one end of the rope is pulled, the load lifts from one position to another.

### Compound Pulley

Compund pulley is a mechanical system in which a fixed pulley and a moveable pulley are combined to work together as one unit and can provide greater mechanical advantage.

## How to find acceleration in a pulley system without friction

Case 1: When M2 moving downwards and M1 moving upwards.

Case 2: When M2 moving upwards and M1 moving downwards.

Case 3: When M2 moving downward and M1 is moving straight on friction less surface.

Case 4: When M2 moving downward and M1 is moving on inclind friction less surface upward at an certain angle θ.

## Conservation of Momentum

The law of conservation of momentum states that when two or more objects collide in an isolated system, the total momentum before and after the collision remains same.

Consider two objects having masses m1 and m2 moving towards each other along the same straight line with velocities u1 and u2 respectively. Let their velocities after the collision are v1 and v2.

By the law of conservation of momentum,

The total momentum of the objects before collision = The total momentum of the objects after collision

m1u1+ m2u2 = m1v1+ m2v2.

This is an universal law because it is valid for the collision of astronomical bodies but also for collision of subatomic particles.

### Applications of Law of Conservation of Linear Momentum

Rocket Launch: In rocket launching, the burning fuel ejects from the lower end of the rocket in downwards direction, which forces the rocket to move in the upwards direction . The mass of the rocket decreases with the burning of the fuel due to which the momentum of the rocket keeps increasing. The total momentum of the system, including rocket and fuel, remains constant as before repulsion of the rocket.

Gun Recoil: When a bullet is fired from gun, the gun puts a force on the bullet to propels it forward. The bullet also exerts an equal and opposite force on the gun in the backward direction. According to Newton’s third law of motion, every action has an equal and opposite reaction. The total momentum of the recoiled gun and bullet remains zero.

## Friction

Friction is a kind of resistance force that opposes the motion of one object against another.

One cause of friction is the irregularities at the microscopic level on the contact surfaces that create interlocking and resisting movement.

Cohesive forces and adhesive forces play an important role in friction between two sliding surfaces.Cohesive forces are the attractions between molecules of the same substance, while adhesive forces are the attractions between molecules of different substances.

### Factors Affecting Friction

Friction is affected by factors by following factors

• the nature of the surfaces in contact.
• the applied force.
• the surface area.
• the smoothness of the surfaces.

Other factors like temperature and the presence of lubricants can also influence friction.

### Effects of Friction

Friction has both positive and negative effects:

1. Positive Effects:

• Friction provides grip between objects. Which helps vehicles to move and pedestrians to walk safely.
• Friction is essential for braking of moving objects to slow down or stop effectively.
• Friction between the pen/pencil and paper helps us to write or draw something.

2. Negative Effects:

• Friction produces heat and causes energy loss in machines and engines, making them less efficient.
• Friction between the surfaces lead to wear and tear of surfaces in contact, reducing their lifespan.
• Excessive friction creates hindrance for the movement of objects, making it harder to slide or roll.

### Type of Friction

There are basically four types of friction:

• Static Friction
• Sliding Friction
• Rolling Friction
• Fluid Friction

Static Friction

Static Friction acts on objects when they are resting on a surface.

Sliding Friction

Sliding Friction acts on objects when they are sliding over a surface.

Rolling Friction

Rolling Friction acts on objects when they are rolling over a surface.

Fluid Friction

Fluid Friction acts on fluid layers when they move.

### Limiting Frictional Force

Limiting friction force is maximum friction force that comes into play when the body just starts moving .

It is the product of the Normal force and coefficient of limiting friction.

It is the maximum value of static friction.

### Laws of Friction

The law of friction states that the. frictional force between two surfaces is directly proportional to the normal force pressing the surfaces together and is also dependent on the nature of the surfaces in contact.

It can be mathematically expressed as F= μ * N where F is the force of friction, μ is the coefficient of friction, and N is the normal force.

The coefficient of friction varies and depending on the materials in contact and can be either static or kinetic, representing friction at rest or in motion, respectively.

### Angle of Repose (θ)

Angle of Repose (θ) is the minimum angle of an inclined plane makes with the horizontal when a body placed on it just begin to slide down.

### Angle of Friction (α)

Angle of Friction (α) is the angle made by the resultant of normal reaction and limiting frictional force with the normal reaction.

### Methods to reduce friction

Friction between the contact susrface can be reduced by adopting following methods.

1. By using lubricants: Lubricants fill the gaps between contact surfaces and reduces friction

2. Polishing surfaces: It reduces the irregularities contact surfaces and hence reduces friction

3. By using ball bearings: Ball bearings to be used r in otating machines parts.

4. By streamlining: streamlining shape reduces the area of contact and thus reduces friction.

## Pseudo-Force

A pseudo force is an apparent force that acts on all bodies whose motion is described using a non-inertial frame of reference,It is also called a fictitious force or an inertial force.

## Uniform Circular Motion

• If an object moves along the circumference of a circle with constant speed is called uniform circular motion.
• The magnitude of velocity of object undergoing uniform circular motion remains constant and its direction changes continuously and direction of velocity is tangential to the circular path.
•  As the object is continuously changing its direction of motion, its velocity changes    continuously. Since the velocity of the object is changing, it has acceleration.
• The acceleration of object moving with a constant speed v in a circular path of radius R & has a magnitude v2/R and it always acts radially and directed towards the canter of the circle. This acceleration is called centripetal acceleration.
• Force that produces acceleration in object is called centripetal force.

Examples of U.C.M

• Motion of the earth around the sun.
• Motion of the moon around the earth.
• Revolution of electron around the nucleus of atom.

## Derivation of centripetal acceleration

Let an object moves on circular path of radius r from point A to B and covers distance Δs  in Δt time.

v1 = velocity at point A
v2 = velocity at point B

Consider triangle ΔPQR  to determine vector difference of velocity v1 & v2.

Δv= (v2-v1)

ΔOBC and ΔPQR are similar triangle .

QR/CB=PQ/OB

Δv/Δr=v/r -----Equation1

Since point A & B are close so Δr=Δs.

Δr=Δs=v.Δt

Putting Δr=v.Δt in Equation 1

=> Δv/v.Δt=v/r

=> Δv/Δt = v.v/r

=> Δv/Δt = v²/r

Δv/Δt is rate change of velocity. Which is equal to acceleration.

centripetal acceleration=v²/r

if mass of object is m then

centripetal force= mv²/r

## Objects in Equilibrium

If an object is at rest or moving with a constant velocity, but it is not accelerating .Then object is said to be in equilibrium.

An object is in equilibrium when all the forces and torques acting upon the object are balanced.

Condition for object in equilibrium , the net external force on the object is equal to equal zero.

## What Is Banking of Roads?

Banking of road is a way of providing the required centripetal force to a vehicle in order to make a safe turn when moving on a curved road.

Banking of road helps vehicle to avoid skidding, overturning or toppling of vehicle.

Consider a vehicle of mass ‘M’ with moving speed ‘v’ on the banked road with radius ‘r’. Let θ be the angle of banking of road, with frictional force F acting between the road and the tyres of the vehicle.

Frictional force between tyres and road =>F= µN

Analysis of forces acting on vehicle:

Frictional force between tyres and road =>F= µN

Components of normal reaction along the vertical axis =Ncosθ

Weight of the vehicle acting vertically downward= Mg

Components of frictional force along the vertical axis= Fsinθ

Components of frictional force along the horizontal axis= Fcosθ

When frictional force is zero the vertical component of the road acts as a normal force due to which the weight of the vehicle is balanced and the horizontal component gives in to the centripetal force in the direction of the Centre of the curvature of the road.

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