Question

show that second law is the measure of force

Answer 1

Let's see what does Second Law of Motion says:-

The rate of change of momentum of a body is directly proportional to the applied unbalanced external force and takes place in the same direction in which the force acts.

Consider a situation in which a car with low battery, is pushed from behind along a straight line, so that it picks up sufficient velocity to start its engine. If one or moreperson give it a sudden push for a very short time, the car hardly starts. However, its results in the increase in acceleration and velocity of Car. From this statement, it implies that Change in Momentum of car is not determined only by the magnitude of Force, but also on the time for which force is applied. Thus, we conclude that Force is necessary to Change the Momentum of a body depend upon the time rate at which the momentum Changes.

We will now prove the Mathematical expression for Newton's Second Law of motion that how it is related to the Force:-

Consider a body os mass 'm' at point A such that its initial velocity is 'u'. Let the body be acted upon by a force 'F' fot time 't' seconds, such that at B, its final velocity is 'v'. Let 'a' be the acceleration of the body during change in velocity.

Initial momentum of the body = mu

Fina momentum of the body = mv

Rate of change in momentum = mv-mu/t

= m(v-u)t

= ma (∴ v-u/t = a)

Now, by Newton's second law of Motion

Rate of Change in momentum ∞ F

ma ∞ F

or

F ∞ ma

F = K. ma  

F = ma (∴ K is a constant)

Let's see how K is constant:-

If there be a body, having a unit mass and unit acceleration, such that the force possessed by it is one unit then,

F = K . ma

1 = K . 1 x 1

∴ K = 1.

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