Newton’s Second Law of Motion

According to Newton’s first law of motion the application of an unbalanced force brings a change in the velocity of an object. Thus, the force can produce a change of momentum. Newton’s second law of motion establishes a relationship between force and change in momentum.

Second law of motion states that the rate of change of momentum of a body is directly proportional to the force acting on it and takes place in the same direction as the force.

Newton's second law of motion also gives a relation between force and acceleration.

Suppose the velocity of an object of mass m changes from u to v in time t by the application of a constant force F.

The magnitude of initial and final momentum of the object will be

p1 = mu

p2 = mv

Change in momentum in time t = p2 - p1

Rate of change of momentum = (p2 - p1)/t

According to second law of motion, the magnitude of the force F, is

F ∝ (p2 - p1)/t

F = k(p2 - p1)/t

where k is constant of proportionality.

F = k(mv - mu)/t

F = km(v - u)/t

F = kma

We choose the unit of force in such a manner that the value of k becomes one. For this, one unit of force as that amount which produces an acceleration of 1 m/s2 in an object of 1 kg mass. So that:

1 unit of force = k (1 kg) × (1 ms–2)

Thus, the value of constant k becomes 1. Therefore,

F = ma

The unit of force is called newton and its symbol is N.

So a force of 1 newton will produce an acceleration of 1 m/s2 on an object of mass 1 kg.

Daily Life Applications

1. While catching a fast moving cricket ball, why does a fielder moves his hands backward?

By doing so the fielder increases the time duration in which the momentum of the ball becomes zero. As the rate of change of momentum decreases, a small force is required for holding the catch. So the hands of the fielder do not get hurt.

2. Why does a person get hurt when he falls on a cemented floor?

Just before touching the floor, the person has some initial velocity, say u, which becomes zero when he comes to rest. Thus the momentum of the person becomes zero within a very short time. As the rate of change of momentum is very high, so very large force is exerted on the person, thereby hurting him. On the other hand, if he falls on sand or husk or on a foam mattresses, he does not get hurt due to longer period of time in making momentum zero and hence reduction of force.

3. How does a karate player breaks a pile of tiles or a slab of ice with a single blow?

The karate player hits the pile of tiles or a slab of ice as fast as possible with his hand. In doing so the entire momentum of the hand is reduced to zero in a very short time. As a result, the force delivered on the tiles or slab of ice is large enough to break it.

Examples

1. What is the acceleration produced by a force of 15 N exerted on an object of mass 3 kg?

According to second law of motion

F = ma

Here, m = 3 kg and F = 15 N

Therefore,

15 N = 3 kg × a

a = 15 N/3 kg

= 5 ms–2

2. What force accelerates a 50 kg mass at 5 ms–2?

Newton’s second law gives

F = ma

Here, m = 50 kg and a = 5 ms–2

Therefore,

F = 50 kg × 5 ms–2

= 250 N