## Newton’s Second Law of Motion

Newton’s first law of motion implies that force produces acceleration in a body. The second law of Newton relates force to measurable quantities like acceleration and mass. The rate of change of momentum of a body is directly proportional to the net force acting on it and takes place in the direction of the net force.

**Illustration 1:** Push a tennis ball. It will produce a small acceleration and require a small velocity in a certain time. If we push the same tennis ball a little hard then it will produce a large acceleration and also it acquires a large velocity in the same interval of time. Therefore, the magnitude of two forces can be compared by measuring the accelerations produced by them at the same time.

**Graphical Representation**

This experiment by Newton concludes that

If force F_{1 }produces an acceleration of 10 m/s

And force F_{2} produces acceleration of 20 m/s on the same object,

Magnitude of force F_{2} = 2 magnitude of F_{1}

**Illustration 2:** Take a tennis ball and a football which are initially at rest. Apply the same change of velocity in the same time to both balls, we need to apply more force on the football as compared to the tennis ball.

Therefore, the force needed to produce the same acceleration in both balls if masses are different. Force for football is more than the tennis ball.

**Example:** If force F is required to produce an acceleration of 5 m/s^{2 }in an object of mass 2 kg, then to produce the same acceleration in another object of mass 4 kg, a force of magnitude 2F is required.

**Graphical Representation**

F ∝ m if acceleration remains same

After combining both the equations, the new relation is

F ∝ m a

F = Kma where K is a constant

The unit of force is chosen by taking K = 1 when m = 1 and a = 1. Thus, the amount of force that when applied on a body of unit mass produces a unit acceleration in the body is taken as one unit of force. With the above unit of force, the equation is as follows.

**Mathematical Expression of Newton’s Second Law of Motion**

Force = Mass × acceleration

F = ma

**Unit of Force**

As we know,

F = ma.

Therefore,

F = kg m/s^{2}.

S.I. Unit of Force = Newton

One newton is the force which when acts on an object of mass 1 kg, produces an acceleration of 1 m/s^{2}

i.e., 1 Newton = 1 kg × 1 m/s^{2}

Newton is represented as N.

C.G.S. Unit of Force = Dyne

One dyne is the force which when acts on an object of mass 1 g, produces an acceleration of 1 cm/s2

i.e., 1 Dyne = 1 g × 1 cm/s^{2}.

**Relationship** **Between Newton and Dyne**

1 Newton = 1 kg × 1 m/s^{2}

= 1000 g × 100 cm/s^{2}

= 10^{5} g cm/s^{2}

= 10^{5} Dyne

**Newton’s Second Law of Motion in Terms of Rate of Change of Momentum**

When a force is applied on a body, it produces an acceleration in the body, because of which, the velocity and thereafter the momentum of the body changes.

Therefore, \(\frac{\Delta p}{\Delta t}=m a\), if mass remains constant

Newton’s second law of motion states

F = m a

Therefore, force = rate of change of momentum

F = \(\frac{\Delta p}{\Delta t}\)

= \(\frac{m \Delta v}{\Delta t}=ma\), if mass remains constant

The rate of change of momentum of an object is directly proportional to the force applied on it and the change in momentum takes place in the direction in which the force is applied.

As we have studied F = \(\frac{\Delta p}{\Delta t}=\frac{\Delta(m v)}{\Delta t}\)

This equation shows that momentum changes because of the change in mass or change in velocity or because of the change in both mass and velocity.

It is seen that the mass of an object increases with the increase in velocity when the velocity v of an object is comparable with the speed of light c (3 × 108 m/s), but at velocities v << c, the change in mass is not perceptible.

As such velocities (v << c), mass m can be considered to be a constant.

Then, Newton’s second law of motion is described as

F = \(\frac{\Delta v}{\Delta t}=m a\)

Thus, relation F = \(m \frac{\Delta v}{\Delta t}\) = ma holds in two conditions:

- When velocities are much smaller than the velocity of light.
- When mass remains constant.

**Relationship between First Law of Motion and Second Law of Motion**

From Newton’s second law F = m a

If F = 0 then a = 0

It means when there is no force applied then acceleration will be zero i.e., if the object is at rest, it will continue to be at rest and if it is moving, it will remain moving in the same direction with the same speed. This is what Newton’s first law of motion states.

**Example:** A boy throws a bottle on the floor and it breaks, but when he throws the same bottle on the carpet it doesn’t break. The reason is when the bottle falls on the hard floor, it comes to rest in a very short time due to which the floor exerts a large force on the bottle, and as a result it breaks.

But when the bottle falls on a carpet the time duration in which the bottle comes to rest increases, therefore carpet exerts less force on the bottle and as a result, it doesn’t break.