Question

Write Newton's second law of motion and prove F = ma from this law. Also define S.l. unit of force "Newton" with the help of this relation and find the dimensional formula of force.

Answer 1

According to this law, "Rate of change of momentum of a moving body is directly proportional to force applied on the body and in same direction in which force is applied.

Suppose the momentum of a moving body is \(\vec p\), then rate of change of momentum \( = \frac{d\vec p}{dt}\)

∴ According to second law of motion, the force acting on the body, 

\(\vec F\propto\frac{d\vec p}{dt}\)

or \(\vec F = K\frac{d\vec p}{dt}\ ...(1)\) 

where K is constant of proportionality. The value of K depends on the system of chosen unit. We chose the units Such that

K = 1

\(\therefore\ \vec F = \frac{d\vec p}{dt}\ ...(2)\) 

Second law of motion is known as real law.

 

Therefore the force applied on a body is defined by the product of mass of body and acceleration produced.

Case (ii) When Velocity is constant, then 

\(\frac{dv}{dt} = 0\) 

\(\therefore\ \vec F =\vec v\frac{dm}{dt}\)

Unit of force : ∵ F = ma

∴ In M.K.S. system, unit of F = kg.ms^-2 = Newton (N)

In C.G.S. system, unit of F= g.cm.s^-2 = dyne

Dimensions: ∵ F = ma

∴ Dimensional formula of F = [M¹L¹T^-2]

Relation between N and dyne:

\(1\ N = 1 \ Kg\times\frac{1m}{1s^2} = \frac{1000g\times100cm}{s^2}\) 

= 10⁵ g.cm.s^-2

or 1 N = 10⁵ dyne

Definition of N: ∵ F= ma

If m = 1 kg; a = 1 ms^-2; then F = 1 N

Thus, “1 N is the force which can produce an acceleration of 1 ms^-2 in an object of mass 1 kg.”