Newton's Second Law of Motion

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Earlier we have mentioned that application of force can change the velocity of the moving object. Let us suppose that a body of mass 'm' is moving with a velocity 'u'. When a force F is applied on the body for 't' seconds, the velocity changes from the initial velocity 'u' to the final velocity 'v'. Hence,

Initial momentum = mu …(i)

Final momentum = mv …(ii)

Change in momentum = mv - mu

This change has been brought about by application of force F for t seconds. Hence, the rate of change of momentum

= (mv - mu) t

= m(v - u) t

This leads us to Newton's second law of motion. According to it,

The rate of change of momentum of a body is proportional to the impressed force and takes place in the direction of the force acting.

According to this law, the impressed force is proportional to the rate of change of momentum. Hence,

But, (acceleration)

Hence,

Therefore, F = Kma, where K is the constant of proportionality. It we choose the unit of force in such a way that it produces unit acceleration on unit mass, then

1 = K x 1 x 1

Therefore, K = 1

Therefore, F = ma

Like velocity and acceleration, force is also a vector quantity as it possesses both magnitude as well as direction. Thus, Newton's first law of motion defines inertia and his second law of motion defines force as mass times acceleration. In other words, the first law gives the qualitative definition of force and the second law gives the quantitative or operational definition of force.

Units of Force

The unit of force in the S.I. is 'newton'.

(Symbol : N)

One newton is that force which when acting on a body of mass 1 kg produces a uniform acceleration of 1 m s^-2 in its own direction.

The C.G.S unit of force is 'dyne'.

One dyne is that force which when acting on a body of mass 1 g produces a uniform acceleration of 1 cm s^-2 in its own direction.

Relationship between Newton and dyne

1 newton = 1kg x 1 m s^-2

= 1000 g x 100 cm s^-2 [F = ma]

= 100000 g cm s^-2

= 10⁵ dyne

As discussed earlier, the relationship between force, mass and acceleration is expressed by the equation,

F = ma

Now if no force acts on a body (F = 0)

0 = ma

But m, being mass of the body can never be zero,

a = 0

Thus, when there is no force acting on a body, no change takes place in its velocity. When the body is at rest it continues to be in that state of rest and if the body is in motion it continues to move with uniform speed along a straight line.