# Basic Physics lesson - 13 : Simple Harmonic Motion

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## Introduction

If we recall, we have studied and learnt about linear movement of bodies in lesson-2 : Speed and velocity, about circular motion of bodies in lesson-10 : Circular motion, and about rotational motion of bodies in lesson-12 : Rotational motion. Now it is is time to go ahead and learn about periodic motion or one of the types of periodic motions known as Simple Harmonic Motion (SHM). In this article we will try to learn about the basic concepts of Simple Harmonic Motion.

## Periodic motion

Periodic motion is defined as a motion of a body which repeats itself after the same time and that time is known as the period or periodic time. Periodic motion can be divided in two broad categories - first is where the force does not reverse it's direction and second is where it reverses it's direction. The examples of first are circular and rotational motions while example of latter are oscillatory or vibratory or Simple Harmonic Motion, in whom, the force reverses its direction and acts in opposite direction of the displacement. If the body follows a linear path in this to and fro motion then it is called linear simple harmonic motion and if it follows an arc of a circle then it is called circular simple harmonic motion.

## Linear Simple Harmonic Motion

This is the simplest of all and can be understood easily. Let us consider a simple movement of a body in a line from point A to B and stop at B and then return back to A and then repeat this cycle (Figure-1). Interesting thing is we can visualise it happening in a particular way like - the body starts it's journey from A under an applied force and acquires some acceleration and gains some velocity by the time it reaches midway at point O. Now it has to stop at point B and that implies that the force on it would reverse as soon as it reaches O and then it would decelerate and reach point B and at that point its velocity will be zero and then the same force would start moving it to point O by accelersting it. So this cycle will go on and the body will oscillate like this till the force is there.

It is obvious that in such a scheme, the body would have highest velocity in the mid point O and lowest at the ends that is A and B. What about acceleration? It would be minimum at the midpoint and highest at the ends.

We have to note that whenever there is a restoring force which is increasing with the displacement of the body then simple harmonic motion is generated. In case of mechanical springs, this is understood in terms of well known Hooke's Law.

## Circular SHM - Learning from a pendulum

Have you seen a pendulum? It is nothing but a small body tied with one end of a string and other end of the string is fastened on a nail in the wall at some height. Let the body hang freely with that string. Now if we displace the body to one side it would start oscillating from one side to another till that initial energy remains in it as imparted by the force. The body follows a path defined by the small circular arc of radius equal to the length of the string. Please note that there are two forces acting on the body one is that we applied on it with that push and other is the weight of the body acting downwards to Earth's surface. Under these two forces the body is having oscillatory motion and it is nothing but simple harmonic motion.

Let us consider the case of a simple pendulum where we have hanged a small ball of mass m from a string of length L (Figure-2). Let us give a push to it to one side and observe what happens. It goes to some distance and stops and starts coming back and then with its momentum it goes to other side and comes back. So it is now doing these oscillations till the initial energy is sufficient to move it and after some time it would eventually stop and come to its normal hanging position. Let us assume that while giving a push, it goes from point A to B and stops and starts coming back. At this point B let us assume the angle between the string and the verticle is θ. The weight mg of the ball is acting downwards but there is a component of it which forces it back to its normal position and that component is mg sinθ. Though we will not go in the mathematical details of derivations but considering the equation of motion of this ball and also assuming that angle of displacement is small enough allowing us to make an approximation of sinθ = θ, we get the formula for time period of one oscillation (T) as -

T = 2π/ω where ω is the natural frequency of this oscillatory motion.

In this case ω is given by (g/L)1/2

Combining above two formulas, we get -

T = 2π (L/g)1/2

This is the most general equation of SHM for the time period T of these oscillations.

### A small exercise

Let us do one exercise where a small ball is hanged with a 1.5 metre length light weightless string and we give it a push to one side and now we want to find out the time period of oscillation. We can use the formula T = 2π (L/g)1/2 and get T = 2 x 3.14 x (1.5/9.8)1/2 and on solving we would get T = 2.4 second approximately.

One interesting thing to note here is that this time period is not affected by the weight of the ball. Weight of the ball is only to provide restoring force and time period depends on the length of the string.

### A simple experiment to find the value of g

This oscillatory motion is very interesting and one can find the value of g by using a simple practical in one's house itself. What we require is a light synthetic string of length from 1 metre to 1.5 metre, one small ball, and a stopwatch. One can also use some timer or stopwatch app from paystore in place of a physical stopwatch. Now fix one end of the string to the ball with some adhesive glue and hang it from a nail on the wall. See that string or ball does not touch the wall. Now push the ball to one side so that it starts oscillating. Note down time for 10 oscillations by observing the ball manually. Repeat this a few times. Your readings would be very near to each other and take their average value and divide by 10 to get the time period of one oscillation. Measure the length of the string in metres. As we know that T = 2π (L/g)1/2 , from this you can easily get the value of g (acceleration due to gravity) at your place. The average value of g on Earth's surface is 9.81 m/s2. You will get a value very near to that. This value slightly changes as per latitude and height of the place where one lives but that difference is very small and we can ignore that for this simple practical experiment.

## Simple Harmonic Motion

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## Conclusion

Simple Harmonic Motion is one type of oscillatory motion evident in many activities happening around us like the vibrational motion of a spring, motion of a pendulum etc. These motions happen under the presence of a restoring force which varies as per the displacement of the body from its normal position.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

Umesh Chandra Bhatt (author) from Kharghar, Navi Mumbai, India on November 04, 2020:

Dale, thanks a lot.

Dale Anderson from The High Seas on November 03, 2020:

Another good physics article here. We all appreciate you sharing these with us.

Umesh Chandra Bhatt (author) from Kharghar, Navi Mumbai, India on October 31, 2020:

Eman, thanks for visiting. Appreciate much.

Eman Abdallah Kamel from Egypt on October 31, 2020:

Thank you for this interesting and informative article.

Umesh Chandra Bhatt (author) from Kharghar, Navi Mumbai, India on October 23, 2020:

Bill, thanks a lot for your polite consideration. Feel encouraged. Stay blessed.

Bill Holland from Olympia, WA on October 22, 2020:

Your articles are always informational and interesting, and for that I thank you!

Umesh Chandra Bhatt (author) from Kharghar, Navi Mumbai, India on October 22, 2020:

Eric, thanks a lot for your lovely comment. You are always humble and kind hearted.

Umesh Chandra Bhatt (author) from Kharghar, Navi Mumbai, India on October 22, 2020:

Liz, thanks a lot for your visit. Appreciate much.

Eric Dierker from Spring Valley, CA. U.S.A. on October 22, 2020:

I must admit that I did not get all of this. But I learned a lot. Thanks

Liz Westwood from UK on October 22, 2020:

This is an interesting and well-presented physics lesson. Much easier to grasp than when I was at school.