# Basic Physics Lesson-12 : Rotational Motion

*Umesh is a freelance writer contributing his creative writings on varied subjects in various knowledge and educational sites in internet.*

## Introduction

We have discussed and learned about circular motion (lesson-10) in our earlier lesson in this basic Physics series. We had seen there the example of Earth moving around the Sun in an approximately circular path. But Earth is also rotating around its own axis. This axis of Earth is almost in North to South line through its centre. It is making one rotation around its axis in one day time. This rotation around its own axis is in fact an example of rotational motion.

Sometimes, students confuse between circular motion and rotational motion. These two are actually different. In the former a body moves in a circle and completes one circular round coming back to same position while in the latter case it rotates around a pivot point. For example if we push a door, its motion around the hinges is a rotational motion. In this case pivot point is at the hinges. Now, in this article, we would try to understand and learn about the rotational motion in details.

## Vote

Earth rotation and our calendarsOur Calendars are based on the fact that Earth completes 1 rotation around its axis in 1 day, divided in 24 hours and also completes 1 round of Sun in 1 year, divided in12 months, each month having 30/31 days. To bring the calendar in tune with the movement of Earth around Sun, February is made of only 28 days. Some error remains which is corrected by making February every 4th year as 29 days. Still a fine error remains and that is corrected by making February of every 400th year as 28.

## What is Rotational motion

Children go to the children's park and enjoy merry-go-round. It rotates around its axis. Children also play with spinning tops which rotate for quite some time. While rotating on the floor they also move here and there in a zig-zag path. So, a rotational body can move in a path also like Moon which is also rotating around its axis and also moving in an almost circular path around Earth. Most of the bodies in the universe are having this type of combined movement.

When a body rotates around itself then it takes some time in taking one rotation. That value of time gives an idea about its rotational speed. If a body is rotating around its axis about 3 times in a minute then it is said to have RPM (Rotation per minute) value as equal to 3. The blade in a domestic mixer grinder rotates with high RPM of the order of 20000. Our car engine has about 1000 rpm speed while idling. The seconds hand in a wall clock takes 1 minute in completing one rotation.

## Kinetic energy of rotating body

When a body of mass m is moving with a constant velocity v then its kinetic energy is defined as mv^{2}/2. In analogy to that, the kinetic energy of the rotation can also be expressed in similar form using the angular velocity (ω).

The linear velocity v and angular velocity ω are related as v = rω, where r is the distance from the axis of rotation. If we put this value of v in the general kinetic energy expression then we get the kinetic energy of rotation as mr^{2}ω^{2}/2, which can also be written as Iω^{2}/2, where I = mr^{2 }

and is known as moment of inertia about which we would learn in details in this lesson itself at a later stage.

So, if a body is moving with a linear velocity and as well rotating also then it's total kinetic energy would be the sum of the above two kinds that is (mv^{2}/2) + (Iω^{2}/2). If you see a drum or cylindrical body or even a ball rotating and moving down a slope then you can visualise this very easily.

## Torque and angular acceleration

When a force is applied on the wheel of a bike it starts rotating around the axle. When a force is applied on a merry-go-round in the playground it starts rotating around its central pipe. So force results in angular motion and from this change of angle we can find out angular acceleration. If we recall our earlier lesson in this series that is linear acceleration (lesson - 3), we found there that an equation of the form F = ma existed where F is the force applied on mass m giving rise to linear acceleration a. In the case of rotational motion also we have the similar relation holding true. Note that angular acceleration and linear accelerations are related to the perpendicular distance (r) of force to the axle or pivot point. The relation is a = rα, where α is angular acceleration.

Now, taking analogy from linear motion, we can present the relationship between force and angular acceleration as -

F = mrα ................. (1)

One interesting thing which we can practically observe is that it is easier to rotate a body by applying a force on the peripheral side rather than in between or near the axle. Have you ever used a leaf cutter for manually cutting the offshoots of bushes in your garden. Our hands are far from the axle while cutting edge is very near to the leaf. We are able to impart good force to it from distance. Many levers are based on these principles. This gives us knowledge of an important entity called Torque. More the torque better it would be. Torque applied at a point is defined as force multiplied by its perpendicular distance from that point. If F is the force and r the perpendicular distance then the torque (τ) is given by -

τ = Fr .................... (2)

We can use the above two equations to get the rotational motion equation as -

τ = mr^{2}α

This equation tells us that torque is creating angular acceleration and how much it would be will depend upon the entity mr^{2} which is known as the rotational inertia or moment of inertia. So we have found a new thing now in hand called moment of inertia and the concept of moment of inertia is very important in understanding the dynamics of rotating bodies.

## Ease of rotation

In many machines, rotating parts are there. The drum in a washing machine, ceiling fan blades in our rooms, wheels in a car, are all rotating parts. Technology is always searching newer ways to design the machines so that the moving parts move easily and freely with minimal consumption of energy. The overall efficiency of a machine largely depends on the effectiveness of the rotating parts having minimal frictional losses and cosequently having less wear and tear.

That is the reason, why rotating parts are kept supported on ball bearings or other bearings which give minimal friction to their movement. If eletrical connections are required from the rotating coils wound on a frame or rotar then we use soft spring loaded metalic bushes for electrical connections which make an electrical path but do not create much mechanical friction. So, wherever there is a rotational motion, necessity of creating ease of rotation is felt.

## Moment of inertia

While learning about torque we found that torque and angular acceleration of a body are related to each other through moment of inertia. We can say it in slightly different way that when a force is applied on a body then its moment of inertia opposes it but finally gives in when the force is sufficient to rotate it. This has a practical utility in finding the limit of force in civil constructions which is otherwise capable to bend a construction and may render it useless. This knowledge is crucial and if we have it, we can keep necessary margins while designing a girdle or any such entity which is sensitive to large bending forces.

The moment of inertia of a rigid body can be considered as comprising of the moment of inertia of individual small masses that constitute the body. By multiplying the each individual mass by the square of its distance from the axis of rotation and adding all of them, we can get the moment of inertia of the whole body around that axis of rotation. We would not go in the mathematical details of those derivations but the basic thing is that there are methods of integration (adding together in a range) by which moment of inertia of a body can be found out.

Just an example here - The moment of inertia of a solid cylinder of mass M and radius R around the axis of the cylinder is MR^{2}/2 while for a hollow cylinder of same mass and size it is MR^{2}. This is very interesting to note that in the solid cylinder the mass is distributed evenly while in the hollow cylinder mass is away from the axis and in between there is no mass, as it is hollow. That is the reason why moment of inertia for hollow cylinder is more than the solid one. Some students and readers might ask how these cylinders are constructed having same mass in those two fashions. The answer is that these could be different materials and the solid might be made with less density material while hollow is made with high density material, something like Aluminium and Steel. This example illustrates a very important thing about moment of inertia and that says that moment of inertia depends on the distribution of the mass around the axis of rotation and this is an important learning that is used in designing rotating parts like flywheels in some of the equipments and machineries.

## Moment of inertia

## Vote

## Conclusion

Understanding of rotational motion of bodies gives insight about the dynamics of the rotations. Their moment of inertia plays an important role as how the bodies would behave when an external torque is applied on them. All these concepts are useful in the industry where designing and manufacturing of rotating parts of various machines and equipments is undertaken.

## References

2. https://www.khanacademy.org/science/physics/torque-angular-momentum

3. https://en.wikipedia.org/wiki/Rotation_around_a_fixed_axis

4. https://www.youtube.com/watch?v=fmXFWi-WfyU

*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**

## Comments

**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:

I fancy myself as a bit of a garage-scientist so I am really enjoying your physics series of hub articles quite a bit.

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

Jay, thanks for the information. I might come back to you once I go through all that information.

**Jay C OBrien** from Houston, TX USA on October 05, 2020:

Recently discovered was the collision of the Dwarf galaxy Sagitarius with the Milky Way Galaxy. The earth is located at the impact site. The Dwarf Galaxy Sagitarius then began looping around the Milky Way several times. The earth area was hit several times over millions of years. It is possible that gravitational forces or debris could have changed the earths' orbit. Look it up.

**Umesh Chandra Bhatt (author)** from Kharghar, Navi Mumbai, India on October 03, 2020:

Jay, thanks for going through the article so meticulously. Calenders are based on the one revolution of Earth around the Sun and if there is even a small mismatch, correction is required. In Indian culture also earlier clenders were having 360 days but after a few years (say 5-6 years) we were adding a blank month to bring it to the correct position. This might be true in other cultures also.

As regards to your query about spin reversal, there is no scientific basis for that that it happened but yes, such reversals can happen if some big interaction with a huge celetial body happens which is moving near the Earth and exchanging huge momentum with it and then disappearing in space without colliding. Very unlikely and rare in the cosmos but theoretically, yes.

**Jay C OBrien** from Houston, TX USA on October 03, 2020:

It is my understanding that ancient calendars had 360 days, not 365 days. This was based on the orbit of the earth around the sun. Calendars were changed to reflect the 365 day cycle. Did the earth slow down at some point in mans' history? Writing on the ceiling of an Egytian tomb state the earth reversed its' spin. Is that possible?

**Umesh Chandra Bhatt (author)** from Kharghar, Navi Mumbai, India on October 03, 2020:

Rajan, thanks a lot.

**Rajan Singh Jolly** from From Mumbai, presently in Jalandhar, INDIA. on October 02, 2020:

Felt like back to school again. Thanks for the lesson.

**Umesh Chandra Bhatt (author)** from Kharghar, Navi Mumbai, India on September 28, 2020:

Lorna, thanks a lot for your beautiful and encouraging comnent on this basic Physics article. Appreciate much.

**Lorna Lamon** on September 28, 2020:

I enjoyed this scholarly trip down memory lane Umesh. I studied Medical Physics at Uni and have always been fascinated by this subject. Thank you for sharing this enjoyable and interesting read.

**Umesh Chandra Bhatt (author)** from Kharghar, Navi Mumbai, India on September 28, 2020:

Flourish, thanks for sparing time for this. Really appreciate.

**FlourishAnyway** from USA on September 28, 2020:

Thank you for this refresher.