Newton’s Laws of Motion

Newton’s 1st law or Law of Inertia

Every body continues to be in its state of rest or of uniform motion until and unless and until it is compelled by an external force to change its state of rest or of uniform motion. 

Inertia 

The property by virtue of which a body opposes any change in its state of rest or of uniform motion is known as inertia. Greater the mass of the body greater is the inertia. That is mass is the measure of the inertia of the body. 
Numerical Application If, F = 0 ; u = constant ( In the absence of external applied force velocity of body remains unchanged.) 

Physical Application of inertia or newtons’s first law

1. When a moving bus suddenly stops, passenger’s head gets jerked in the forward direction.
2. When a stationery bus suddenly starts moving passenger’s head gets jerked in the backward direction.
3. On hitting used mattress by a stick, dust particles come out of it.
4. In order to catch a moving bus safely we must run forward in the direction of motion of bus.
5. Whenever it is required to jump off a moving bus, we must always run for a short distance after jumping on road to prevent us from falling in the forward direction. 

Newton’s Second law of motion :

It states that the rate of change of momentum of a body is proportional to the applied force and takes place in the direction in which force acts. 

Thus F= k dp/dt= k ma 

Derivation of second laws of motion 

⇒ The second law is consistent with the First law (F=0 implies a=0)
⇒ It is a vector equation
⇒ It is applicable to a particle, and also to a body or a system of particles, provided F the total external force on the system and a is the acceleration of the system as a whole. 
⇒ F at a point at a certain instant determines acceleration at the same point at that instant.Acceleration at an instant does not depend on the history of motion’ 
⇒ Force is not always in the direction of motion .Depending on the situation F may balong V, opposite to v, normal to v, or may make some other angle with v. In everycase it is parallel to acceleration. 
⇒ If v=0 at an instant, i.e., if a body is momentarily at rest, it does not mean that force or acceleration are necessarily zero at that instant. For ex: When a ball thrown upward reaches its maximum height, but the force continues to be its weight ‘mg‘ and the acceleration is ‘g’ the acceleration due to gravity. 

Note :- Above result is not Newton’s second law rather it is the conditional result obtained from it, under the condition when m = constant. 

Numerical Application: 

acceleration, a = F _Net / M

Where F _Net is the vector resultant of all the forces acting on the body

Physical Application 

i) Case - 1 Body kept on horizontal plane is at rest.  
For vertical direction 
N = mg(since body is at rest) 

ii) Body kept on horizontal plane is accelerating horizontally under single horizontal force. 
For vertical direction 
N = mg (since body is at rest) 
For horizontal direction
F = ma 

iii) Body kept on horizontal plane is accelerating horizontally towards right under two horizontal forces. (F1> F2) 
For vertical direction
N = mg (since body is at rest) 
For horizontal direction F1 - F2= ma 

Tension 

Tension In A Light String Force applied by any linear object such as string, rope, chain, rod etc. is known as it’s tension. Since string is a highly flexible object so it can only pull the object and can never push. Hence tension of the string always acts away from the body to which it is attached irrespective of the direction. 

Physical Application of Tension 

i) Flexible wire holding the lamp pulls the lamp in upward direction and pulls the point of suspension in the downward direction.

ii) Rope holding the bucket in the well pulls the bucket in the upward direction and the pulley in the downward direction.

iii) Rope attached between the cattle and the peg pulls the cattle towards the peg and peg towards the cattle.

iv) When a block is pulled by the chain, the chain pulls the block in forward direction and the person holding the chain in reverse direction. 
In case of light string, rope, chain, rod etc. tension is same all along their lengths. 

Consider a point P on a light (massless) string. Let tensions on either side of it be T₁ and T₂ respectively and the string be accelerating towards left under these forces. Then for point P2

Since string is considered to be light mass m of point P is zero or, T = 0 

or, T₁ - T₂ = ma 

T₁= T₂

Tension of A light Rigid Rod

Force applied by rod is also known as its tension. Since rod is rigid, it cannot bend like string. Hence rod can pull as well as push. Tension of rod can be of pulling as well as pushing nature but one at a time. Tension of a rod attached to the body may be directed towards as well as away from the body. 

Physical Application of tension :-

i) Pillars supporting the house pushes the house in the upward direction and pushes the ground in the downward direction.

ii) Wooden bars used in the chair pushes the ground in the downward direction and pushes the seating top in the upward direction.

iii) Parallel bars attached to the ice-cream trolley pushes the trolley in the forward direction and pushes the ice-cream vendor in the backward direction.(when the trolley is being pushed by the vendor) 

iv) Rod holding the ceiling fan pulls the fan in the upward direction and pulls the hook attached to the ceiling in the downward direction. 

v) Parallel rods attached between the cart and the bull pulls the cart in the forward direction and pulls the bull in the backward direction. 

Fixed Pulley

It is a simple machine in the form of a circular disc or rim supported by spokes having groove at its periphery. It is free to rotate about an axis passing through its center and perpendicular to its plane. 

Key Point

In case of light pulley, tension in the rope on both the sides of the pulley is same (to be proved in the rotational mechanics)

Newton’ 3rd law or Law of Action and Reaction

Every action is opposed by an equal and opposite reaction.
or 
For every action there is an equal and opposite reaction. 

F₁₂ is the force on the first body (m₁ ) due to second body (m₂ )

F₂₁ is the force on the second body (m₂) due to first body (m₁) 

If F₁₂ is action then F₂₁ reaction and if F₂₁is action then F₁₂ reaction.

Numerical Application 

Force on the first body due to second body (F₁₂ ) is equal and opposite to the force on the second body due to first body (F₂₁ ).

F₂₁ = - F₁₂

Physical Application 

i) When we push any block in the forward direction then block pushes us in the backward direction with an equal and opposite force. 

ii) Horse pulls the rod attached to the cart in the forward direction and the tension of the rod pulls the cart in the backward direction.

iii) Earth pulls the body on its surface in vertically downward direction and the body pulls the earth with the same force in vertically upward direction.

iv) While walking we push the ground in the backward direction using static frictional force and the ground pushes us in the forward direction using static frictional force. 

v) When a person sitting on the horse whips the horse and horse suddenly accelerates, the saddle on the back of the horse pushes the person in the forward direction using static frictional force and the person pushes the saddle in the backward direction using static frictional force. 

Note – Normal reaction of the horizontal surface on the body is not the reaction of the weight of the body because weight of the body is the force with which earth attracts the body towards its center, hence its reaction must be the force with which body attracts earth towards it.