Newton's Second and Third Law of Motion

Newton’s Second Law of Motion

Newton’s Second Law of Motion states that the rate of change of momentum of a body with respect time is directly proportional to the net external force applied on it and the change takes place in the direction of force.

When a force F is applied on a body whose momentum at time t is p, then from Newton’s second law we have,

$$\begin{align*} F &\propto \frac {dp}{dt} \\ \text {or,} \: F &= k \frac {dp}{dt} \\ \end{align*}$$

Where k is a constant of proportionally, whose value depends upon the choice of unit of force. The system of units are so chosen that k = 1.

$$\therefore F = \frac {dp}{dt} $$

If m is the mass and v is the velocity of the moving object, the linear momentum, p = mv and

$$\begin{align*} F &= \frac {d}{dt} (mv) \\ &= m \frac {dv}{dt} \\ &= ma \\ \text {as m is constant for a given body.} \\ \therefore F &= ma \end{align*}$$

Above equation gives the magnitude of the force if we know the acceleration of the object.

SI-Units of Force

The SI-units of mass is kg and that of acceleration is ms^-2.

$$\begin{align*} \therefore \text {SI-unit of force}, F &= 1 kg \times 1ms^{-2} \\ &= 1 kg ms^{-2} \\ &= 1 \text {Newton} &= 1N \\ \end{align*}$$

So, is an acceleration of 1m sec^-2 is produced on an object of mass of 1 kg, then the magnitude of force is called 1 Newton. The unit of force in CGS-system is dyne.

Newton’s Third Law of Motion

Newton’s third law states that, “To every action there is equal and opposite reaction.” This law shows that force occur on pair. But we must know that action and reaction do not cancel each other as the action and reaction force do not act on same body but act on two different bodies. Some examples of third law of motion are

1. Swimming
   While swimming in the pond, the swimmer pushes the water in backward direction where as the reaction of the water pushes the swimmer in the forward direction. So the swimmer can swim easily in the water.
2. Flight of a rocket
   The burnt fuel come out of the rocket in the form of hot and highly compressed gases in the backward direction whereas the escaping gases push the rocket in the forward direction.
3. Walking
   When we are walking on the road, we push the ground in backward direction, the man gets reaction force in forward direction.

Second Law is the real Law of Motion

Second law contains both first and second law. That is why it is called the real law. If no external force is applied on the object i.e. F = 0, then ma= 0 or a = 0 as m≠0. So, u = constant. That means an object at rest will remain at rest and an object at rest will remain at rest and an object moving with uniform speed on a straight line will continue to do so, which is the first law.

To prove that the third law of motion is also contained in the second law of motion, we take two bodies A and B moving along the same straight line. As a result of collision, their momentum will change. Let t be the time of impact, then change in linear momentum of A is

$$\begin{align*} \Delta p_A & = F_A \times t \\ \text {Similarly change in linear momentum of B is} \\ \Delta p_B & = F_B \times t \\ \text {So the total change in linear momentum A and B is} \\ \Delta p &= \Delta p_A + \Delta p_B = F_A \times t + F_B \times t \\ \text {If no external force acts on the system, then} \Delta P = 0 \\ \text {or,} \: F_A \times t + F_B \times t &= 0 \\ \text {or,} \: F_A &= -F_B \text {which is Newton’s third law of motion.} \\ \end{align*}$$