The Force Law For Simple Harmonic Motion

The external force is responsible for the motion of a body to move back and forth at a fixed interval of time. This external force is always proportional to the displacement made by the body. The relation which relates this external force to the displacement of the body is nothing but Hooke’s law. The law gave certain conditions to be followed by external force considering the simple harmonic motion. The external force that acts on the body is the restoring force opposite to the direction of displacement of the body.

Let us now briefly go through and understand the simple harmonic motion.

Simple harmonic motion

A simple harmonic motion is a periodic motion that repeats itself after some time. SHM can be considered to perform oscillatory motion if it moves to and fro. A simple pendulum would be a simple and real-life example that shows simple harmonic motion. All the examples that perform to and fro motion are simple harmonic motions. 

All oscillatory motion can be considered periodic, but the opposite is not always true.

To understand the force law acting in simple harmonic motion, one needs to know about Hooke’s law, also known as the law of elasticity.

Hooke’s law:

Hooke’s law is also called the law of elasticity. Robert Hooke, an English scientist, discovered it in the middle of the 16th century. Hooke’s law states that the force applied to the string is directly proportional to its extension, provided it does not pass the elastic limit.

If an elastic body follows Hooke’s law, the object returns to its original shape and size upon removal of the force. 

The behaviour of elastic solids can be explained easily using Hooke’s law because small displacements of their constituents from normal positions are also proportional to the force that causes the displacement.

Mathematically, Hooke’s law can be shown as,

F= -kx

Here, F is the restoring force that is deforming the elastic body,

K is the constant, and x is the displacement made by the body.

Consider a vertical spring attached to the ceiling that has a mass ‘m’. When we consider the case where we push the body connected with the vertical spring towards the ceiling, the spring contracts, achieving a displacement of +x, but the force acting on the body would be just opposite to it in a downward direction.

Similarly, in another case, when it is left loose to stretch, the body would experience an external force( tension due to spring) upwards. In contrast, the displacement would be this time downwards, which is opposite to the external force acting on the body.

In this way, we can generalise the relationship between the external force and the displacement of the body, deforming the elastic body as a system.

Force law for simple harmonic motion

To understand the force law in SHM, let us take an example, a simple spring block system, where we consider a system of a block of mass ‘m’ attached with a spring. The spring is connected to the wall. 

Suppose, at first, the block is at its equilibrium position. 

We will perceive the two cases with the help of the spring block system to understand what restoring force is.

Now try to imagine pulling the block towards the outward direction; the external force due to spring would try to bring it back to its equilibrium position. Similarly, if the block is pulled towards the inward direction, this time as well, the external force would try to push the block back to its equilibrium position.

In both scenarios, external force tries to push back the block to its equilibrium position. 

You see, an external force acts about the mean position of the body. This external force is nothing but the restoring force, and this is the force law for SHM. The restoring force is responsible for restoring the body to its equilibrium position.

Mathematically, there are two forms of this restoring force:

F= -kx

This is the simplest form of force law in SHM. Here, F is the restoring force, k is the force constant, and x is the displacement of the body. The negative sign shows that the restoring force acting on a body is always opposite in direction to the displacement of the body.

Another form of force law in SHM can be derived from Newton’s second law, 

 F=ma

 a= –kmx

a= –2x

This is the second form of describing the force law in SHM.

Conclusion

Simple harmonic motion is a periodic motion that repeats after a fixed time. A simple pendulum would be a good example that shows simple harmonic motion. All these motions are related to the external force that a body experiences. There is a law called the law of elasticity or Hooke’s law to understand this external force. The law explains the restoring force, which is responsible for bringing back the body to its mean position. Mathematically, the force law can be understood by different forms of the force law using Newton’s second law of motion.