Question

State Newton's second law of motion. Hence, derive the equation of motion $F=m.a$. From it obtain the unit of force in SI. Show that Newton's second law of motion is the real law of motion.

Answer 1

Step 1: Given.

Newton's second law of motion

Step 2: Explanation.

We know that Newton's second law, the acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object, i.e.$a=\frac{\overrightarrow{F}}{m}$. The second law states that the rate of change of momentum of a body is directly proportional to the force applied, and this change in momentum takes place in the direction of the applied force.
$F=\frac{dp}{dt}$
Momentum, $p=m(v-u)$
$F=m\frac{d(v-u)}{dt}$
As we know acceleration is defined as $a=\frac{v-u}{t}$
So, $F=ma$
A unit of force is ${\mathrm{kgms}}^{-2}$ as a unit of mass is $\mathrm{kg}$ and a unit of acceleration is $m{\mathrm{s}}^{-2}$.
Newton's second law of motion is real law: This can be proven if the first and third law of motion can be described by the second law of motion.
As $F=ma$,
It shows that there will be no acceleration in the body if there is no external force applied i.e. a body in motion will be in motion if no external force is applied and similarly a body at rest will remain at rest if no force is applied. Hence first law is contained in the second law. Now for the third law.

Consider an isolated system of two bodies A and B acting and reacting internally.

$F_{AB}^{}=$ force applied on body A by body B
$F_{BA}=$ the force applied on body B by body A
and $\frac{d{p}_A}{dt}=$ rate of change of momentum of body A
$\frac{d{p}_B}{dt}=$ rate of change of momentum of body B
So, from Newton's 2nd law of motion, ${ϝ}_{AB}=\frac{d{p}_A}{dt}\cdots \left(i\right)$ and
$F_{BA}=\frac{d{p}_B}{dt}\cdots \left(ii\right)$

From the (i) and (ii), $F_{AB}+F_{BA}=\frac{d{p}_A}{dt}+\frac{d{p}_B}{dt}=\frac{d}{dt}(p_A$ $\left.+p_B\right)$

As the system is isolated, no external force acts on the system, so $\frac{d}{dt}\left(p_A+p_B\right)=0$
Hence, $F_{AB}+F_{BA}=0$
or$F_{AB}^{}=-F_{BA}$
This is Newton's 3 rd law derived using Newton's 2nd law. As we have seen Newton's 1 st and 3rd laws both are contained in the 2nd law, hence it is the real law of motion.