Question

Prove that Force= mass $\times$ acceleration. State the condition when it holds.

Answer 1

Newton's second law says that force is the rate of change of momentum. The acceleration of the object is directly proportional to the force and inversely proportional to the mass.

Complete step by step answer:
According to Newton's second law of motion, The rate of change of momentum of a body is directly proportional to its force. If $\overrightarrow p$ the body's momentum, m is the body's mass and v is the body's velocity, then momentum is given below.
$\overrightarrow p = m\overrightarrow v$
Now let us write Newton's second law of motion.
$\overrightarrow F \propto \dfrac{{d\overrightarrow p }}{{dt}}$
Here, F is the force.
Now, let us substitute the value of $\overrightarrow p$, and we get the following.
$\overrightarrow F \propto \dfrac{{d\left( {m\overrightarrow v } \right)}}{{dt}}$
Now, as we know mass m is constant then we can write the expression as follows:
$\overrightarrow F \propto m\dfrac{{d\left( {\overrightarrow v } \right)}}{{dt}}$
Again, we can rewrite the above expression as below:
$\overrightarrow F \propto m\overrightarrow a$
Here $\overrightarrow a$ is the acceleration is given by $\overrightarrow a = \dfrac{{d\overrightarrow v }}{{dt}}$ which means that the rate of change of velocity is called acceleration.
We can replace the proportionality sign introducing a constant k,
$\overrightarrow F = km\overrightarrow a$
If the proportionality constant k=1, then the above equation becomes,
$\overrightarrow F = m\overrightarrow a$
Here, F is the force and has the unit of newton, m is the mass of unit kg and a is the acceleration of unit ${\rm{m/}}{{\rm{s}}^{\rm{2}}}$.
The above relation holds where the body's mass is constant, and the velocity of the body is very less than the velocity of light.

Note: To derive F=ma, we consider mass to be constant and the proportionality constant to be unity. When we study the theory of relativity where an object's speed is nearly equal to the speed of light, then F=ma does not apply. As the higher speed is almost identical to the speed of light, mass no longer remains constant; it becomes variable.