Consequences of Newton’s Second Law of Motion

The following are the direct consequences of Newton’s second law of motion:

1. Concept of inertial mass.

From Newton’s second law of motion, it follows that

$ a = \dfrac{F}{M} $

i.e., the magnitude of the acceleration produced by a given force is inversely proportional to the mass of the body. More is the mass of the body, lesser will be the acceleration produced. Thus, the mass of the body is the measure of the resistance offered by the body to the change in velocity which the applied force tends to produce i.e., mass of a body is the measure of its inertia. For this reason, mass given by the above equation (Newton’s second law of motion) is called  inertial mass.

2. An accelerated motion is always due to a force.

The accelerated motion of a body can occur in the following three ways:

(a) Due to change in its speed only. The force must be acting on the body along the direction of motion or opposite to the direction of motion.

(b) Due to change in direction of motion only. The force must be acting perpendicular to the direction of motion of the body. Such a force makes the body move along a circular path and is called centripetal force.

(c) Due to change in both speed and direction of motion. The force must be acting obliquely i.e., at some angle with the direction of motion. The component of force along the direction of motion produces a change in speed, while that along the normal produces a change in direction.

3. The measurement of applied force.

According to Newton’s second law of motion,

$$ F = M \cdot a $$

If the inertial mass of the body is known, then by measuring the acceleration produced, the magnitude of the force applied can be determined.

Suppose that the velocity of the body changes by $\Delta v$, when force $F$ acts on the body for a small time $\Delta t$. Then, acceleration is given by:

$$ a = \frac{\Delta v}{\Delta t} $$

and therefore the applied force is given by

$$ F = M \cdot \frac{\Delta v}{\Delta t} $$

If the inertial mass of the body is known, then by measuring the change in velocity produced and the time for which the force acts on the body, the magnitude of the force can be found.

4. The acceleration produced by a force in the motion of a body depends only upon its mass.

The value of the acceleration produced by a force acting on a body does not depend upon the velocity it had in the past or upon its present velocity. It depends only upon the mass of the body.

5. The applied force can be measured from the force law applicable to the mechanism.

In an accelerated motion, the applied force can be measured from the knowledge of the force law applicable in that mechanism.

For example, consider a body attached to a spring. If we pull the spring with some force, the body attached to the spring will move with some acceleration and the length of the spring will also increase. In this mechanism, the fact that the increase in length of the spring is directly proportional to the force applied is the ‘force law’.

Suppose that $ \displaystyle \Delta {{l}_{\circ }}$ is increase in lenght of the spring, when tha body attached to the spring is pulled with a force of 1N. If the body is pulled with an unknown force F, due to which the length of the spring increases by $ \displaystyle \Delta l$, then the force applied (in N) is given by

$ \displaystyle F=\dfrac{{\Delta l}}{{\Delta {{l}_{\circ }}}}$

A spring balance used to measure the weight of a body is based on this principle.

6. The slope of momentum ($ \displaystyle p$) versus time ($ \displaystyle t$) curve gives us the force, i.e.

$ \displaystyle \text{Slope of }p-t\text{ curve }=\dfrac{{dp}}{{dt}}=F$