Laws of Motion Chapter - Physics JEE Main

Name of the Concept	Key Points of Concept
1. Newton's Law of Motion	Newton's First Law of Motion Newton's First Law of Motion, also known as the law of inertia, states that an object at rest will stay at rest, and an object in motion will stay in motion with the same speed and in the same direction unless acted upon by an unbalanced external force. The key points to remember are: Inertia: Every object has a natural tendency to resist changes in its state of motion. No net force = No acceleration. Newton's Second Law of Motion Newton's Second Law of Motion relates the net force acting on an object to its acceleration and mass. Mathematically, it can be expressed as F = ma, where: F is the net force applied to an object. m is the mass of the object. a is the acceleration produced by the force. This law provides a quantitative relationship between force, mass, and acceleration Newton's Third Law of Motion Newton's Third Law of Motion states that for every action, there is an equal and opposite reaction. It means that when one object exerts a force on a second object, the second object exerts an equal and opposite force on the first. This law explains how interaction between objects occurs.
2. Linear Momentum and Impulse	Linear Momentum is defined as the product of mass and linear velocity, $\vec{P}=m v$ Impulse force acting on a body is defined as the product of force and the infinitesimally small interval of time for which it acts, $I=\int_{t_1}^{t_2}\vec {F} dt=\vec {P_2}-\vec{P_1}$
3. Conservation of Linear Momentum	The law of conservation of linear momentum states that “when external forces are not acting, then total momentum of the body remains constant.”  In other words, the momentum of the body before impact will be the same as the after impact. Mathematically we write: ${m_1}{v_1}={m_2}{v_2}\Rightarrow {v_1}=\dfrac {{m_2}{v_2}}{m_1}$
4. Concurrent Forces and Equilibrium	When a group of forces will be acting on a single point, such forces are known as the concurrent forces If the resultant force of these forces is zero, then these concurrent forces are said to be in “equilibrium”.
5. Resultant Force	Resultant force is the vector sum of all the forces acting on the body.
6. Friction	Friction is the opposition offered when any object is moving a plane. The formula of the friction force is: $\vec  {F}=\mu N$ Relation between the coefficient of friction and of friction $\theta$: $\theta=\tan^{-1}\mu$
7. Laws of Friction	The total magnitude of limiting frictional force is always proportional to the normal force acting at the contact surface. The magnitude of limiting frictional force is always independent of the total surface area of contact between the surfaces.
8. Tension and Pulley System	Tension is defined as a force acting on a stretched string or body.  A pulley system is a basic physical apparatus with two or more masses attached to it.
9. Calculation of Tension Acting On a Body	The direction of the tension acting on the string is always considered along the string and in a direction opposite to the weight of the body under consideration. $\Rightarrow T=m g$ A block Attached to a String
10. Calculation of Tension In a Simple Pulley	The common acceleration ‘a’ acting in a simple pulley system: $a=\dfrac {{m_1}-{m_2}}{{m_1}+{m_2}} g$ The tension in the string is given by $T=\dfrac {2{m_1}{m_2}}{{m_1}+{m_2}} g$ Where, m1>m2

Examples of Newton’s Law of Motion in Daily Life

Newton’s First Law of Motion:

• A small coin is put on a card and placed over a glass. When the card is flicked away with the finger, the coin drops into the glass.

• Suppose we are sitting in a stationary bus. If it starts moving suddenly, we will feel a jerk in the backward direction. It is because our lower body is in contact with the seat of the bus that comes in motion as the bus starts moving, while the upper portion of our body remains at the rest due to inertia, and so we feel a jerk in the backward direction.

• Similarly, if we push a ball on the ground, it should continue its uniform motion indefinitely, but it stops after covering a certain distance. As soon as the ball starts moving, a force (force of friction) comes into play, which opposes the motion of the ball.

• A book lying on a table can't change its position by itself unless a force is applied to change its position.

Newton’s Second Law of Motion:

• Pushing or pulling an empty cart is easy as compared to a loaded cart because the loaded cart has more mass.

• If the same amount of force is applied to move a car and a bike, the acceleration of the bike will be more because it has less mass than the car.

• When a ball drops on the ground, it exerts a downward force on the ground, and in reaction to it the ground exerts an equal upward force on the ball, thus making it bounce.

• Stopping a moving ball requires force.

Newton’s Third Law of Motion:

• When a man or woman is out on a stroll, he/she exerts force in a backward direction on the ground. On the other hand, the ground puts a forward directional force on the man/woman that makes him/her move forward.

• Another Newton's 3rd law example is when an individual is swimming, he/she pushes the pool water backward. In return, the water pushes the swimmer in a forward direction, which makes him/her proceed.

• Agreeing with Newton's third law, a bird pushes air downwards with the help of its wings. Conversely, the air helps the bird to move in the upward direction.

Derivation of First Equation of Motion

Let’s Consider a body of mass m having initial velocity u.

Let after time be t its final velocity becomes v due to uniform acceleration a.

Now it is defined as:

Acceleration = Change in velocity / Time taken

Acceleration = (Final velocity - Initial velocity) / Time taken

$ a = \dfrac{(v - u)}{t}$

$ a t = v - u$

or  $v = u + at$

This describes the first equation of motion.

Derivation of Second Equation of Motion

As it is defined, the Second equation of motion: $s = ut + \dfrac{1}{2}at^2$

Let’s take the distance travelled by the body be s.

Now:

Distance = Average velocity x Time

Also, Average velocity $= \dfrac{(u + v)}{2}$

Therefore, Distance $(t) = \dfrac{(u + v)}{2t}$       ......eq.(1)

Again from first equation of motion:

$v = u + at$

Substituting this value of v in eq.(1), we get

$s = \dfrac{(u + u + at)}{2t}$

$s = \dfrac{(2u + at)}{2t}$

$s = \dfrac{(2ut + at^2) }{2}$

$s = \dfrac{2ut}{2} + \dfrac{at^2}{2}$

or  $s = ut + \dfrac{1}{2} at^2$

This describes the second equation of motion.

Derivation of Third Equation of Motion

As it is defined that the third equation of Motion:  v2 = u2 + 2as

Now,

v = u + at

v - u = at

or  $t = \dfrac{(v - u)}{a}$            ........ eq.(2)

Also ,

Distance = average velocity x Time

Therefore, $s = \dfrac{v + u}{2} \times \dfrac{v - u}{a}$

$s = \dfrac{(v^2 - u^2)}{2a}$

$2as = v^2 - u^2$

or  $v^2 = u^2 + 2as$

This describes the third equation of motion.

Inertia:

Inertia is the property of a body due to which it opposes any change in its state. The mass of a body is the measure of its inertia of translational motion. It is difficult to change the state of rest or uniform motion of a body of heavier mass and vice-versa. In simple words, inertia is a measure of the tendency of an object at rest to stay at rest or of an object in motion to stay in motion.

• Mass of a body is a quantitative or numerical measure of a body's inertia.

• Larger the inertia of a body, the more will be its mass. 

The Inertia of Rest 

A body cannot change its state of rest by itself.

Examples:

• When we shake a branch of a mango tree, the mangoes fall. 

• When a bus or train starts suddenly, the passengers sitting inside tend to fall backward.

• When a horse starts suddenly, the rider falls backwards.

• The dust particles in a blanket fall off when it is beaten with a stick.

• A coin is placed on cardboard, and this cardboard is placed over a tumbler, such that the coin is above the mouth of the tumbler. Now, if the cardboard is removed with a sudden jerk, the coin falls into the tumbler.

The Inertia of Motion 

A body cannot change its state of uniform motion by itself.

Examples:

• When a bus or train stops suddenly, the passengers sitting lean forward.

• A person who jumps out of a moving train may fall in the forward direction.

• A bowler runs the ball before throwing it so that this speed of running gets added to the speed of the ball at the time of the throw.

• An athlete runs through a certain distance before taking a long jump because the velocity acquired during the running gets added to the velocity of an athlete at the time of the jump, and hence he can jump over a longer distance.

• A ball is thrown in the upward direction by a passenger sitting inside a moving train.

The ball will fall:-

• back to the hands of the passenger, if the train is moving with constant velocity. 

• ahead of the passenger, if the train is retarding (slowing down).

• behind the passenger, if the train is accelerating (speeding up).

The Inertia of Direction 

A body cannot change its direction of motion by itself. The tendency of an object to oppose any change (unbalanced force) in its direction of motion and continue to stay in direction is called inertia of direction.

 Examples:

• When a straight running car turns sharply, the person sitting inside feels a force radially outwards.

• Rotating wheels of vehicles throw out mud, mudguards fitted over the wheels prevent this mud from spreading.

• When a knife is pressed against a grinding stone, the sparks produced move in the tangential direction. 

What is an External Force?

According to Newton's first law of motion, a push or a pull that either changes or tends to change the state of rest or uniform motion (constant velocity) of a body is known as a force. An external force is a force originating from outside an object rather than a force internal to an object. 

For instance, the force of gravity that Earth exerts on the moon is an external force on the moon. However, the force of gravity that the inner core of the moon exerts on the outer crust of the moon is an internal force on the moon. Internal forces within an object can't cause a change in that object's overall motion.

Effects of Resultant Force     

• It may change the speed of the body.     

• It may change the direction of motion.     

• It may change both the speed and direction of motion.     

•  It may change the size or/and the shape of the body.      

• It may start a motion in a stationary body or it may stop a moving body.

Units for Measurement of Force

Absolute units are-  

• N (M.K.S)     

• dyne (C.G.S)

Other units are-    

• kg-wt or kg-f {kf-force}    

• g-wt or g-f

Acceleration and Velocity

Newton's second law of motion describes that, when a force is applied to an object, it produces acceleration in the object (i.e rate of change of velocity). For an object at rest, the applied force produces acceleration in the object and makes the object move in the direction of applied force.

For an object which is already in motion, the direction of the applied force matters to determine its state. If external force is applied in the direction in which the object is moving, the acceleration of the object increases. If external force is applied in the direction opposite to the motion of the object, the acceleration of the object decreases and finally comes to stop.

Force and acceleration are vector quantities, i.e they have both magnitude and direction. Multiple forces can also act in a body at a time.

Hence,

\[\Sigma\]F = ma

Where;

\[\Sigma\]= vector sum of all the forces acting on a body (net force).

For Changing the Mass

For this let’s assume that we have a car at a point (0) which is defined by the location X0 and time t0. The car has a mass of m0 and travels with a velocity of v0. Here, after being subjected to a force ‘F’, the car starts to move to point 1, defined by location X1 and the time by t1. The mass and the velocity of the car changes during the travel to values m1 and v1 respectively. Thus, Newton's second law helps to determine the new values of m1 and v1 if we already know the value of the acting force.

From the difference of point 1 and point 0, an equation for the force acting on the car is formed as follows:

F=m1v1−m0v0t1−t0

F=m1v1−m0v0t1−t0

Now, let’s assume the mass to be constant here. This assumption is helpful for a car as the only change in the mass would be the fuel burned while moving between point “1” and point “0”. Here, the weight of the fuel is probably very small as compared to the rest of the car, especially looking at the small changes in time. Meanwhile, if we discuss the flight of a bottle rocket, then the mass does not remain constant here, and only the changes in momentum can be looked at.

For Constant Mass

For a constant mass, Newton’s second law can be equated as follows:

F=mv1−v0t1−t0

F=mv1−v0t1−t0

We know, acceleration is defined as the change in velocity which is divided by the change in time.

The second law then decreases to a more common form as follows:

F=ma

The above equation conveys to us that an object will accelerate if it is subjected to an external force. While the amount of force is directly proportional to the acceleration and inversely proportional to the object’s mass.

Law of Conservation of Linear Momentum:

The law of conservation of linear momentum states that the total linear momentum of a system of objects remains constant unless an external force acts on the system. Linear momentum is defined as the product of mass and velocity (p = mv).

The law of conservation of linear momentum can be applied to a variety of problems, including:

• Collisions between objects

• Explosions

• Rocket motion

• Motion of objects in a gravitational field

Applications of the Law of Conservation of Linear Momentum:

• Collisions: When two objects collide, the total linear momentum of the system before the collision is equal to the total linear momentum of the system after the collision. This principle is used in the design of airbags and other safety devices.

• Explosions: When an object explodes, the total linear momentum of the system before the explosion is equal to the total linear momentum of the system after the explosion. This principle is used in the design of explosives and rockets.

• Rocket motion: Rockets work by expelling mass at high speed. The law of conservation of linear momentum tells us that the total linear momentum of the rocket system (rocket + exhaust gases) must be constant. This means that the rocket will accelerate in the opposite direction to the exhaust gases.

• Motion of objects in a gravitational field: When an object falls under gravity, its linear momentum increases. This is because the force of gravity is acting on the object. The law of conservation of linear momentum tells us that the object's total linear momentum must be constant. This means that the object's velocity must increase as it falls.

Equilibrium of Concurrent Forces:

Concurrent forces are forces that act at the same point on an object. An object is in equilibrium when the net force acting on it is zero. This means that the vector sum of all the forces acting on the object must be zero.

You can explore comprehensive details regarding the equilibrium of concurrent forces to get a more straightforward explanation.

Static and Kinetic Friction:

• Friction is a force that opposes the relative motion of two surfaces in contact. There are two types of friction: static friction and kinetic friction.

• Static friction is the force that prevents two surfaces from sliding against each other. It is the force that keeps your feet from slipping when you walk.

• Kinetic friction is the force that opposes the sliding motion of two surfaces in contact. It is the force that makes your brakes work.

Explore the Important Concepts with Solved Examples of Static and Kinetic Friction to gain a deeper understanding of these fundamental principles in physics.

Rolling Friction:

Rolling friction is a type of friction that occurs when a wheel rolls over a surface. It is caused by the deformation of the surface and the wheel. Rolling friction is much smaller than static or kinetic friction.

Applications of Rolling Friction:

Rolling friction is used in a variety of applications, including:

• Wheels

• Bearings

• Ball bearings

Dynamics of Uniform Circular Motion:

Uniform circular motion is the motion of an object that moves in a circle at a constant speed. The centripetal force is the force that keeps the object moving in a circle. The centripetal force is directed towards the centre of the circle.

Applications of Centripetal Force:

Centripetal force is used in a variety of applications, including:

• Banks on roads

• Artificial satellites

• Washing machines

• Centrifuges

To understand the concept of Dynamics of Uniform Circular Motion, one must delve into the principles governing motion in a circular path.