# Basic Physics Lesson-10 : Circular Motion

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## Introduction

We have learned in earlier chapters in this series of lectures about displacement (lesson-1) of the objects, their velocities (lesson-2), and their acceleration (lesson-3). So far these objects were moving in a straight line which we call linear motion. So, by using the basic equations of motion we could find the distance covered by these objects in a given time.

Now, in this lecture we would learn about circular motion. When we talk about the circular motion then it is the simplest one out of the various curvilinear motions happening around us. If we want that a stone from one point on Earth surface should reach to another point then we throw it in the air with some angle by imparting some force on it and then it goes in a curvilinear path to reach that point. The path travelled by the stone is not a circular path but is known as parabolic one. What happens is that under the force which we imparted on it and the Earth's gravitational field it moves in a parabolic trajectory. Now if we tie this stone with a fixed length string and rotate it over our head by holding the other end of the string in our fist then the path taken by the stone is circular motion.

## Background discussion

Before going to learn the physics of circular motion it would be better to visualise some of the circular motions happening around us and the easiest that comes to my mind is the revolving of Earth around Sun.

Most of us are aware that Earth revolves around the Sun and it takes 1 year time to make one complete round. It is also curvilinear motion and precisely speaking it is more near to eliptical motion. For sake of simplicity and ease of rough calculations we take it like circular motion only as these two are the closest cousins when it comes to curvilinear motion.

Sun being the centre of our solar system all the planets revolve around Sun in more and less eliptical paths and complete their rounds around Sun in different different times depending upon their relative mass and distance from the Sun. In addition to the planets, comets also revolve around Sun albeit their revolution time is very large sometimes in thousands year also and they revolve around Sun in highly elongated eliptical paths and when they pass around Sun due to Sun light and heat they glow spectacularly. Seeing a comet in the night sky is always a thrill.

The Moon is revolving around the Earth in almost a circular path and as it is near to us, as compared to planets and stars, we are seeing it so clear as a disc in the night sky which is of course due to reflection of sunlight from it.

Let us come to Earth itself which is rotating around its own axis once a day and simultaneously moving around Sun in annual cycles. As we are living on Earth we do not feel this motion. To visualise the motion of a body one should be away from it. An observer on Moon or elsewhere in space can easily see Earth moving across the sky from that particular place.

So, as we are dwelling on Earth we all are moving along it in its orbit around the Sun. It is like a train carrying a person from one place to another and he or she does not move and sits comfortably inside. He would feel himself stationary but an observer from outside will easily see him passing by. So we conceive the movement of things as per the frame of reference where we are actually present.

## Revolution and angular displacement

Let us first try to understand the concept of revolution in a circular motion. When an object is moving in a circular path with a constant velocity then it takes some time in making one round or one revolution in completing that circular path. Depending on the velocity the time of revolution is less or more.

For example, let us see the case of a person who is going for a walk in the park where there is a circular pathway of length (the circumference of the circle) 300 meters. Let us assume that he walks at a speed of 5 km/hour. How to find out the revolution time. Let us try as follows.

He is walking 5000 metres in 60 minutes.

So he would walk 1 metre in 60/5000 minutes.

Hence he would walk 300 metres in (60 x 300)/5000 = 3.6 minutes that is the time taken in one round of the park. So the revolution time of the person in that circular path is 3.6 minutes. We would be denoting this time as T in this lesson. In the above example value of T was found as 3.6 minutes.

Let us now understand angular displacement. When a body is moving in a circular path then it moves actually along the circumference of that circle. While doing so it goes from one point to another and in doing so an angle is formed at the centre of the circle. This angle is actually the angular displacement over that time. Let us understand it from the diagram-1. The object moves from point A to B in a time t with a velocity v. Angle formed at the centre point O is actually angle AOB and we can denote it by θ (pronounced as theta) for our convenience as well as established nomenclature. If we know the velocity of the object, time taken from A to B, radius r of the circle that is OA or OB, then we can find out θ.

Let us now recollect the basic relationship between the length of a part of the circumference of a circle (known as an arc) and the radius of the circle. The relationship is given by -

Which is nothing but -

θ = AB/r

Scroll to Continue

This is a very important relationship and is used for calculations in a circular motion. If we recollect we will find that the circumference of a circle is 2π times its radius. Here π is a constant and equal to about 3.14.

Another interesting thing is that when you look at π as an angle then it is equal to 180 degrees. Now just see that the angle made by a half-circle at the centre is 180 degrees only. What about the full circle? How much angle it makes at the centre. Is not it 360 degrees? From the same analogy, the quarter circle would make an angle of 90 degrees at the centre. So, what I want to emphasise is that π is also a way to represent angles other than degrees.

This leads us to a new unit of angle that is radian and that is what we use in the calculations in circular motion problem-solving. So to understand the concept we would observe that the whole circumference 2πr makes an angle of 360 degrees at the centre and if we compute it with our formula θ = AB/r then we would get 360 degrees = 2πr/r = 2π.

So, from this simple derivation, we get a very valuable conversion that is 2π radian is equal to 360 degrees and so one radian is equal to 360/6.28 = 57.3 degrees.

Please remember that when you divide AB length with radius r then you get angle θ in radian and not in degrees. To find the value in degrees use the above conversion. This is the most important thing in calculations.

## Angular veocity

When a body is moving in a circular path it has some velocity but the interesting thing to note is that this velocity is changing direction while the object is moving and another interesting thing is that the body comes back to its original position after one round and moves on for next round. Just like the Earth revolving around the Sun and after one year it comes to the same place and almost same alignment with respect to the sun and that is why we have same weather after that one year time. As Earth is inclined on its own axis by an angle of about 23 degrees, the distance from the Sun and how the Earth is facing the Sun basically decides the weather at a place. This would of course also explain why it is winter in Australia when it is summer in Europe and also why it is summer in Australia when it is winter in Europe.

When a body is moving in a circular path with a constant velocity then the magnitude of the velocity is the same but it is changing its direction continuously and after each round comes back to the same direction. There is another way to define this movement and one can find out how much angle it is moving with respect to a reference point. Let us try to understand this aspect with the help of the earlier diagram that is diagram-1. An object is moving in a circular path with a velocity v and while it travels from point A to B in a time t, it makes an angle θ at the centre of the circle and then we define angular velocity ω (pronounced as omega) as -

ω = θ/t

It is a simple relationship that tells us that in a unit of time how much angle the object had moved with respect to the centre of the circle.

## Angular velocity - Exercise

Let us do one exercise for understanding angular velocity. We will try to find out the angular velocity of Earth around Sun. Though the path of Earth orbiting around Sun is slightly elliptical but for convenience of our calculations, we would take it as circular only. We will be using formulas required for these calculations from diagram-1.

We know that Earth takes 1 year to complete a revolution around sun so we have T = 1 year = 365 x 24 x 60 x 60 seconds. The angular displacement is 360 degree or 2π radian.

So, the angular velocity = 2π / (365 x 24 x 60 x 60) = 1.99 x 10-7 radian/second or 1.14 x 10-5 degree/second.

It may look like if Earth is moving so slowly around Sun. It is actually not so. It is the angular velocity and not the simple velocity. We can find the simple velocity also. For that we require the radius of this circular path (Earth's orbit). The average radius of this circular path is about 1.5 x 108 kilometer. Now we can find the velocity by dividing the circumference with time taken to traverse it.

So velocity of Earth = (2π x 1.5 x 108)/(365 x 24 x 60 x60) = 29.9 kilometer per second. In our day to day units of speed that is actually about 107,000 km/h or 67,000 mph. Is not that a huge speed?

## Centripetal and Centrifugal forces

Once we have understood the basic concepts of circular motion, I will like to touch upon as what we understand by centripetal and centrifugal force though the detailed about them would be dealt separately in a future lesson.

When a body is moving in a circular path then it is being actually pulled to the centre with some force which is known as centripetal force. The direction of this force is always towards the centre. So the interesting thing is wherever the body is positioned on the circular route the centripetal force acts inwards on it that is towards the centre. The question that comes in mind is that if it is so why the body does not fall to the centre to honour that force. The answer is that there is an equal and opposite force acting on it outwards and is known as centrifugal force and the body remains in its circular motion.

The formula for centripetal force is given by mv2/r where m is the mass of the body, v is its velocity, and r is the radius of the circular path.

## Conclusion

When we move from linear motion to circular motion then we have to understand some concepts of this movement which are best understood by learning about the angular displacement, angular velocity, revolution time etc and their inter relations.

1. https://www.physicsclassroom.com/class/circles/Lesson-1/Mathematics-of-Circular-Motion

3. https://www.toppr.com/guides/physics/motion/uniform-circular-motion/

4. https://byjus.com/jee/circular-motion/

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.

© 2020 Umesh Chandra Bhatt

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

Dale, thanks a lot for your sweet words.

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

I know I've said it before, but it is worth saying again, I am enjoying your physics articles and hope that you keep writing them.

Umesh Chandra Bhatt (author) from Kharghar, Navi Mumbai, India on September 26, 2020:

Anurag, thanks a lot for your nice comment.

Anurag on September 25, 2020:

Well written article.

Brushed up my basics on the topic!!

A great lecture to read for all physics aspiring people out there.

Looking forward for more lectures ahead.

Umesh Chandra Bhatt (author) from Kharghar, Navi Mumbai, India on September 05, 2020:

CS, thanks for your encouraging comment. Appreciate.

Chitrangada Sharan from New Delhi, India on September 05, 2020:

Great article with wonderful information.

It brought back memories of my physics classes.

Thank you for sharing this well explained article.

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

Eric, thanks for your encouraging comment. Appreciate much.

Eric Dierker from Spring Valley, CA. U.S.A. on August 31, 2020:

I don't think I got it all but I sure learned a whole bunch. Good stuff.

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

Vandana, thanks a lot.

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

Liz, I have prepared it not only for students but also for normal people who have not studied science but are interested in general reading. I do not know how much successful I have been in doing that.

thoughtsprocess from Navsari (India) on August 31, 2020:

Amazingly penned Sir.

Liz Westwood from UK on August 31, 2020:

I appreciate the way that you relate the theory of physics to the world around us. This makes the learning much more accessible to your readers.

Umesh Chandra Bhatt (author) from Kharghar, Navi Mumbai, India on August 30, 2020:

Flourish, thanks and happy to see that it could rekindle memory of your school time. Stay blessed.

FlourishAnyway from USA on August 30, 2020:

Thank you Umesh. This was the part of high school physics when I just wanted to graduate and move on.