Newton's Second Law

We know the effects of unbalanced forces on objects. We have also learned that Newton’s first law talks about inertia. That is, an object will continue to maintain its original position unless it is being disturbed by some unbalanced external force.

Newton’s second law talks about the effects of an unbalanced force on the moving object. The law relates unbalanced force applied to the object with its mass and acceleration produced due to the force.

It states that (unbalanced) force acting on a moving body is directly proportional to the rate of change of momentum of the body in the direction of the applied force. 

This means, if an unbalanced force applied on a moving body is increased then its momentum (amount of motion) will also increase and vice-versa. The change in momentum is in the direction of applied force (Remember: Object always moves in the direction of the greater force).

Explore the following flow chart to understand the dependence and the derivation.

Momentum is another name for the amount of motion contained in a moving body. It is equal to the product of the mass of the moving object and its velocity.

momentum(p)=mass (m)×velocity(v)

Or 

p=mv

From the law, it is clear that:

Force acting on the body (F)∝rate of change of momentum of the body(p)

Or;

F∝p/t……………(1)

Notice that we divide momentum by time to get its rate of change.

We know:

p=mv…………(2)

Substituting equation 2 in equation 1, we get:

F∝mv/t……………(3)

From our earlier lesson we know;

v/t=a……………(4)

Now, substituting equation 4 in equation 3, we get:

F∝ma

That is, introducing a constant ‘k’, we have:

F=k.ma

Where ‘k’ is the constant of proportionality. 

If, the value of;

Mass (m) = 1 kg,

Acceleration (a) = 〖1 ms〗^(-2)  or 1 m⁄s^2 , and

Force (F) = 1N, then the value of constant of proportionality is equal to one (k=1). 

Keeping k=1, we have;

F=ma

Or 

Force (F)=mass(m)×acceleration(a)

This is the mathematical interpretation of Newton’s second law, that is, the force acting on a moving body is equal to the product of the mass of the body and its acceleration.

In order words, the acceleration produced in a moving object because of the unbalanced force depends directly on the magnitude of the force applied and indirectly upon its mass.

The following diagram represents the dependence.

The dependence of acceleration on applied force and mass of the object is discussed below.

Defining one Newton.

The above equation also helps us to define one Newton. (This is one reason why we put value for the constant of proportionality (k) as 1).

We have;

F=ma

That is, If; 

m=1kg

a=1〖m/s〗^2

Then, F=1N

Therefore, one newton can be defined as a force that produces an acceleration of 1〖m/s〗^2 on a body of mass 1kg.

Numerical Problems

We can use the formula to solve numerical problems related to force, mass and acceleration of the moving body. The following are some examples. However, it is important to note that word problems help us to understand the concept clearly and hone your mathematical skills to apply it to real-life situations.

Let us revisit the formula first. We have;

F=ma

Here, force ‘F’ is the subject of the formula. However, as per the requirement for the particular word problem, we can apply the arithmetic operations on to the formula and make either mass or acceleration the subject of the formula. 

The following formula triangle might also help you remember the formulas better.

The position of the physical quantities in the triangle represents its position in the formula. For example, if you want to find the force, the mass should be multiplied by acceleration and if you want to find the mass, force should be divided by acceleration.

That is;

1. F=ma

2. m=F/a

3. a=F/m

Example 1

Calculate the force on a ball of mass 2kg accelerating at the rate of 〖2m/s〗^2.

(Hint: Try using the BUCK method while solving word problems. If you are not familiar with this method, please do explore!)

Solution: 

Given:

Mass (m) = 2kg

Acceleration (a) = 〖2m/s〗^2

We know; 

F=ma

That is; 

Force on the ball (F) = ma

Or;

F=2×2=4N

Therefore, the force on the ball is 4N.

Example 2

A net force of 20N is exerted on a metal ball to cause it to accelerate at a rate of 〖4m/s〗^2. Determine the mass of the ball.

Solution:

Given:

F = 20N

a = 〖4m/s〗^2

We know:

m=F/a

That is;

m=20/4=5Kg

Therefore, the mass of the metal ball is 5kg.