Question

According to Newton’s law of motion correct equation is: (where symbols have their usual meanings)
A) $F = \dfrac{{dP}}{{dt}}$
B) $F = ma$
C) $F = v\dfrac{{dm}}{{dt}}$
D) All of these

Answer 1

Hint: Newton’s second law of motion gives the relation between applied force and momentum of the body.

Complete step by step solution:
As we know that second law of Newton is defined as
The rate of change of linear momentum of a body is directly proportional to the applied force and the change takes place in the direction of the applied force.
$F = \dfrac{{dp}}{{dt}}$ ................. (1)
If a body has mass $m$ moving with velocity $v$ then the linear momentum is
$ \Rightarrow p = mv$
Differentiating both sides with respect to time $t$
$\dfrac{{dp}}{{dt}} = \dfrac{{d\left( {mv} \right)}}{{dt}}$
$\dfrac{{dp}}{{dt}} = m\dfrac{{dv}}{{dt}} + v\dfrac{{dm}}{{dt}}$
From equation (1)
$ \Rightarrow F = m\dfrac{{dv}}{{dt}} + v\dfrac{{dm}}{{dt}}$................. (2)
This equation gives the applied force on body
Most of time in question the mass of body is constant so in that situation $\dfrac{{dm}}{{dt}} = 0$ then above equation (2) becomes
$ \Rightarrow F = m\dfrac{{dv}}{{dt}}$
We know $\dfrac{{dv}}{{dt}} = a$, acceleration of body
$\therefore F = ma$ It is also known as Newton’s law
If the mass of body is varying with time and velocity is constant then $\dfrac{{dv}}{{dt}} = 0$ equation (2) becomes
$F = v\dfrac{{dm}}{{dt}}$, By this we also calculate the force
But the original form of Newton’s second law is $F = \dfrac{{dp}}{{dt}}$

So in this question option (A) is correct.

Note: Students may be confused with other options why $F = ma$ is not correct. As shown in above this formula comes only when mass of the body is constant. Means it is a special case of Newton’s second law in which mass is constant. If mass is varying then this formula is not valid.
But $F = \dfrac{{dp}}{{dt}}$ is always valid. So the answer must be option A.