Newton’s second law of motion

Introduction

Newton’s second law of motion relates to the behaviour of the objects in which all the forces are unbalanced. The law states that the acceleration of the object depends on two variables:

• force on the object and
• the mass of the object

The acceleration and the force are directly proportional to each other. This means that when there is an increase in the applied force, the acceleration also increases. Mass and acceleration are inversely related to each other, meaning that when the mass increases the acceleration decreases.

According to Newton, an object can only accelerate in the presence of net force or unbalanced force. At equilibrium conditions, the object will not accelerate.

Newton’s second law of motion

The formal statement of Newton’s second law of motion is as follows:

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.

This can be expressed as:

a = Fnet / m

Rearranging the equation, we get:

F = m•a

where F = force on the object;

           m = mass of the object;

           a = acceleration.

One unit of force is equal to the unit of the mass multiplied by the unit of the acceleration.

1 Newton = 1 kg • m/s²

This means that one newton is the amount of force required to accelerate the object of mass of 1 kg at an acceleration speed of 1 m/s².

Derivation of Newton’s second law

For deriving Newton’s second law of motion, two cases are defined:

• For constant mass

• For changing mass 

Derivation for constant mass

For a constant mass, the equation for the law can be written as follows:

F = m• (v1-v0) / (t1-t0)

As we know, acceleration is the rate of change in velocity divided by change in time, as per the second law:

F = ma

The above equation states that the object will be accelerated under the influence of an external force. In such a case, the amount of the force will be directly proportional to the acceleration and will be inversely proportional to the mass of the object.

Derivation for changing mass

This can be understood with the help of an example. Let us consider a car at a point ‘0’ at a location X0 and time t0. Now the car has a mass of m0 and is travelling with a velocity of v0.

When the car is subjected to the force F, the car moves to point ‘1’ at location X1 and time t1. Now the velocity will be v1 and mass m1.

Newton’s second law will now be used to determine the values of m1 and v1 (only if we know the value of force acting on the object).

By calculating the distance between points 1 and 0, we get an equation:

F = ( m1v1 – m0v0) / (t1 – t0)

In this case, the mass is assumed to be constant, as the mass change will be the fuel used from point 0 to point 1.

Example: A force of 50 N is applied on an object for 10 sec. Determine the momentum of the object.

Ans: Given force F = 50 N

Time t = 10 sec

We have to find out the momentum of the object.

From Newton’s second law, we know that F = P/t

So that momentum will be

P = F.t

P = 50 N x 10 sec

P = 500 N sec =  50 kg m/sec 

Applications of the second law of motion

We can see the second law of motion applied in these situations.

• Pushing carts: It can be observed that it is easier to push an empty cart rather than a full cart.
• Pushing a truck and a car: When a car and a truck are pushed with the same intensity of forces, the car moves faster than the truck because the mass of the car is less than the mass of the truck.
• Racing cars: It can be noticed that the cars specifically designed for racing purposes have a reduced weight to increase the speed.
• Launching a rocket: When a rocket leaves the Earth’s orbit and approaches outer space, a force referred to as thrust is required. The magnitude of the thrust is increased, which increases the acceleration. This speed finally helps the rocket escape the Earth’s gravitational field and reach outer space.
• While playing tennis: The tennis ball develops some acceleration after it is hit. This acceleration is directly proportional to the force applied to the ball. The ball will move faster if it is hit hard.
• Cycling: The mass is the bicycle, and pushing the pedals by legs is the force. When we push the pedals, the bicycle accelerates, and the harder we push, the greater the speed of the bicycle.

Conclusion

Acceleration is defined as the rate of change in velocity, i.e., a change in either magnitude or direction, or both. According to the law, the acceleration of an object is directly proportional to the magnitude of the force. This means that acceleration increases when the force increases, and acceleration decreases when force decreases. Acceleration is inversely proportional to the mass of the object (if the mass is increased, acceleration decreases and vice-versa).