Simple Harmonic Motion for IIT JEE

We observe different types of motions on a daily basis. Some of these are motion of the car wheels, hands of the clock, etc. There is something interesting to note in these types of motions. It is that they all keep on repeating themselves. Such type of motion is said periodic motion. Let us know more about it.

So, what does it mean by simple harmonic motion?

Let us see the definition of simple harmonic motion. Here a harmonic motion of fixed amplitude is seen where the acceleration is directly proportional to the displacement of a body from equilibrium. The acceleration is in opposite direction to the body’s displacement. The speed of the particle at which it displaces can be different at different instances of time. 

So, how will you characterizea simple harmonic motion?

SHM or a simple hormonic motion is an oscillatory motion. This is a specific case of a periodic motion that is characterized by the following ways:

• A restoring force that has to be applied on the body.

• It is required that the body should have an acceleration in an opposite direction to the displacement.

• Also, the acceleration has to be directly proportionate to the displacement.

Now let us look at the conditions for a simple hormonic motion:

Simple hormonic motion is a kind of a vibratory motion where the body’s acceleration is directly proportional to its displacement. The acceleration is fully directed in the direction of the mean position or the equilibrium point. Following are the basic conditions that are needed to execute a SHM. These are mentioned as:

• There has to be an elastic restoring force to act on the system.

• The system that you are considering must have inertia in it.

• The system’s acceleration should be in proportion to its displacement. It should be always directed towards its mean position.

Oscillations and Periodic Motion

To be able to understand the simple hormonic motion, it is very important to understand the periodic motion. A periodic motion is the one that repeats on its own at equal time periods. It is called as a periodic motion. This is a type of motion where an object repeats on its own path at regular and fixed time intervals. 

We see several examples of such motion in our daily life. The oscillatory motion of the clock is again a periodic motion. The swinging motion on a swing, rocking of a baby on a cradle, to and fro movement of leaves due to wind breeze, are all common examples of a periodic motion.

Putting in other words, here in this type of motion, the particle keeps on performing the same type of movement continuously in the form of a periodic motion. One of such types of movement is referred to as an Oscillation. 

When any object moves and displaces from its position in a “to and fro” manner along a line, then this motion is known as a simple harmonic motion. One of the common examples of it is a pendulum? On swinging, it displaced ‘to and fro’ on the same line. 

This movement is what is called as oscillations and when this movement happens in a repeated manner at fixed time intervals, then it gives rise to a periodic simple hormonic motion. 

To understand more about the simple harmonic motion, we will consider one more example of a spring. This spring is affixed at one corner. When no force is applied to the spring, then it remains fixed at its equilibrium point. When it is pulled in an outward direction, then a pressure is applied by the string. This pressure is exerted towards the equilibrium point. On pushing the spring in the inward direction, a force is exerted by it in the direction of the equilibrium point. 

Harmonic Motion

The force that gets applied by the spring in the direction of the equilibrium / mean position is referred to as the restoring force. Suppose this force is denoted as F and the amount of displacement of a string from the position of an equilibrium position is x, then the restoring force will be F= – kx

Here the force is shown as a negative value that indicates that it is applied in the opposite direction. k is a constant here and it is called as the force constant. The unit of restoring force in S.I. system is N/m and in C.G.S. system it is dynes/cm.

Concepts of SHM or Simple Harmonic Motion

There are a few concepts of simple harmonic motion that you have to know in order to fully understand the concept of simple hormonic motion. 

Amplitude: 

The maximum displacement of any particle from its mean or equilibrium position is called its amplitude. The S.I. unit of amplitude is meter. Its dimensions are “L1M0 T0”. The direction of the amplitude is always opposite to the mean position.

Period: 

The total amount of time required by any particle to make one complete oscillation is called its period. So, in case of S.H.M, it is the least amount of time post which its motion will start to repeat on its own. In other words, the motion of SHM will repeat on itself post nT seconds. Here, n is an integer quantity.

Frequency: 

Frequency of a simple hormonic motion is the total oscillations that a performed by any particle per unit time. The S.I. unit of “frequency” is rotations per second or hertz. The dimension of frequency is L0M0T-1.

Phase: 

Phase of a simple hormonic motion is the state of its oscillation. The direction and magnitude of displacement of any particle representsits phase. Phase of a SHM at the starting of any motion is called as Epoch (α).

Now let us understand how energy is calculated in a simple harmonic motion

Linear SHM or Linear simple harmonic motion

Linear form of simple harmonic motion is described as the linear and periodic motion of a body where the restoring force always directs in the direction of the mean position. Its magnitude is completely proportional to the amount of displacement in its position from the mean position. One important thing that you need to note is that all simple harmonic motions or SHM are periodic by nature, but not all periodic motions are simple harmonic motion.

Now, consider the string example. Let the mass of the string be ‘m’. The total amount of acceleration in the body at any particular instant is given as,

a = F/m 

a = – kx/m 

a = – ω2x

Here ω2 has replaced k/m, (ω denotes angular frequency of a body)

The time period of SHM or simple harmonic motion is independent of energy, phase constant or amplitude.

What are the differences between a simple harmonic motion and a periodic motion?

In case of a periodic motion, the direction of the displacement of an object can or cannot be towards the restoring force. When seen in the case of a simple harmonic motion, the direction of the displacement is always in an opposite direction to the application of the restoring force.

In case of the periodic motion, it can or cannot be oscillatory whereas in case of a simple harmonic motion it is always oscillatory. Examples of a simple hormonic motion are the movement of the hands of a watch, movement of a pendulum, and the movement of a spring, etc.

Simple harmonic motion serves as a fundamental for the classification of several complex motions by way of the methods of the Fourier analysis. When the system gets displaced from its mean position, then a restoring force that follows “Hooke's law” restores the entire system to its equilibrium state. As the body mass starts to get displaced from the equilibrium position, it undergoes restoring force. 

As a consequence, it accelerates and again goes back to its equilibrium point. When the body mass moves nearer to the equilibrium point, then the restoring force starts to decrease. At the equilibrium point, the total restoring force starts to disappear. At point x = 0, the body mass has got momentum due to the acceleration that is provided by the restoring force. Therefore, the body mass continues till the equilibrium point, and compresses the spring. 

The total restoring force slows down till the velocity reaches near 0; there it gets accelerated again to the equilibrium point. Till the entire system does not observe any loss in energy, the body mass keeps on oscillating. Some of the examples in daily life that is a real-world depiction of simple hormonic motion are.

Uniform circular motion

This type of SHM is considered as a 1-dimensional projection of any uniform circular motion. 

Guitar strings

On plucking a guitar string, you will find that it gets displaced at a fixed distance. When it is pulled again to the original position, then it again moves to some distance. 

Bouncing a ball

On bouncing a ball, there is some force applied on the floor. The floor applies the same amount of opposite force towards the ball. So, the ball starts to retaliate towards its starting position. This happens in a repeated manner.