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Simple harmonic motion (S.H.M.)

Learn about simple harmonic motion, its formula, definition, and example. Moreover, learn about the difference between periodic, simple, and harmonic motion.

A simple harmonic motion can be described as a repetitive to and fro motion through the central point/position or equilibrium position. The maximum displacement of the object from one end to the central point is equal to the maximum displacement from the other end to the central point; as a result, the time interval of two vibrations is the same. The restoring force is always directed to the equilibrium position, and it is also directly proportional to the displacement of the object from the central position. This can be represented as F=-kx. In this equation, f is used to represent force and x is used to represent displacement, and k is the constant term. This relationship that we have just studied is known as Hooke’s law. Let us now dig into the details to know more about the topic.

Meaning of simple harmonic motion

Simple harmonic motion definition: Simple harmonic motion is a repetitive to and fro motion through the central equilibrium position. The restoring force is directed towards the equilibrium point, and it is directly proportional to the displacement of the object from the central position.

Simple harmonic motion formula

Let us understand the simple harmonic motion formula using an example. Imagine that spring is placed at a surface. In case of no external force, the spring remains in its resting position, i.e., the mean/central/equilibrium position. When someone pulls the spring outwards, i.e., when the external force is applied, then you will notice that an equivalent force is applied by the spring towards the central equilibrium position (in this case, inwards). When someone pushes the spring inwards, then the force is applied by the spring towards the central position (in this case, outwards). Here, you must have noticed something. Whenever the spring is displaced from its mean position, a force is exerted by the spring directed towards the equilibrium or mean position. Let us now use F to denote force and x to denote the displacement of the spring from the equilibrium position. Thus we can represent the restoring force as: F = -kx In this equation, the negative sign (-) shows that the force is directed towards the opposite direction. The term ‘k’ in this equation is a constant term called the force constant. The unit of this term ‘k’ is Newton per meter. Let the mass of the spring be m. In that case, the acceleration (a) will be: a=Fm a=-kxm a =–𝜔²x By comparing coefficients, km= 𝜔² The time period is the time required by the object to complete one oscillation. The frequency of simple harmonic motion is the total number of oscillations taken by the object per unit of time. Therefore we can represent the frequency as: f = 1/T In the above equations: a = acceleration T= time period F = force f = frequency m = mass 𝜔 = angular frequency k = force constant

Simple harmonic motion examples

To understand simple harmonic motion, we will now take the help of some examples. Here are some objects that undergo a simple harmonic motion:

Pendulum

A pendulum oscillates to and from equilibrium position. It is one of the most common examples of simple harmonic motion.

Swing

Another example of the simple harmonic motion is a swing. It moves back and forth in repetitive movements.

Bungee jumping

Another simple harmonic motion example is bungee jumping. The up and down oscillation of the jumper is an example of SHM because of the elasticity of the cord.

Hearing

Another simple harmonic example is the process of hearing. When the sound waves enter our ears, it makes our eardrums vibrate back and forth.

Cradle

A single push to the cradle causes it to move back and forth. The movement is maintained because of simple harmonic motion.

Simple harmonic motion types

Simple harmonic motion or SHM can be divided into two types:

Angular SHM
Linear SHM

Angular SHM or angular simple harmonic motion

When an object oscillates angularly with respect to an axis, it is known as angular SHM or angular simple harmonic motion.

Some conditions required for angular SHM

Angular acceleration or restoring torque acting on the object must be proportional to the angular displacement of that object and directed towards the equilibrium position. α ∝ θ or θ ∝ T In this equation, θ = angular displacement α = angular acceleration T = torque

Linear SHM or linear simple harmonic motion

An object moves back and forth about an equilibrium position along a straight line. In that case, the motion is known as linear SHM or linear simple harmonic motion. An example of linear SHM is a spring-mass system.

Some conditions required for linear simple harmonic motion

Acceleration or the restoring force acting on the object must be in proportion with the displacement of that object, and it must be directed towards the equilibrium position.

Periodic, simple, and oscillatory motion

Students often get confused between periodic, simple, and harmonic motion. In this part of the article, we are going to differentiate these three motions so that the concept becomes crystal clear.

Periodic motion

In periodic motion, there may not be an equilibrium position
Restoring force is absent
The motion is repetitive. It repeats after a fixed interval of time like the uniform circular motion
The stable equilibrium position is absent