The Ratio Of Rotational And Translational Kinetic Energy Of A Sphere: Detailed Explanations


This article discusses about the ratio of rotational and translational kinetic energy of a sphere. The ratio represents a situation where the sphere is rolling without slipping on a horizontal plane.

When a sphere rolls on the plane, there are two things which can take place. They are- rolling with slipping and rolling without slipping. When the sphere rolls but slips entire without having any translational kinetic energy, then the sphere does not move forward. We shall study more about it in later sections of this article.

What is the ratio of rotational and translational kinetic energy?

The rotational kinetic energy represents the kinetic energy with which the sphere is rotating. Translational kinetic energy represents the kinetic energy with which the sphere is moving forward.

When we divide rotational and translational kinetic energies,  the resulting ratio gives us an idea about a special condition called as rolling without slipping. As discussed in the earlier section rolling without slipping is a condition in which the sphere has two types of kinetic energies- Translational kinetic energy and rotational kinetic energy.

How to find rotational kinetic energy from translational kinetic energy?

The translational kinetic energy is given by the formula – KE= 0.5 mv^2

Rotational kinetic energy is given by the formula- RKE= 0.5 Iw^2

To find rotational kinetic energy from translational kinetic energy we follow the steps given below-

  • First we convert linear velocity to angular velocity by the formula- V = W.r
  • Then we convert mass to moment of inertia by multiplying square of radius of gyration with the mass.
  • This way we have moment of inertia, I and angular velocity, W, so now we can find rotational kinetic energy.

Describe the relationship between rotational kinetic energy and translational kinetic energy

In this section we shall discuss how both of these kinetic energies are related to each other. The only difference between them being the type of motion followed by the object. So let us see the relationship between them in the section below.

Rotational kinetic energy is proportional to moment of inertia which is rotational analogous to mass. Likewise the translational kinetic energy is also proportional to mass. Also, they both are proportional to square of velocity and angular velocity respectively.

the ratio of rotational and translational kinetic energy of a sphere
Image: Kinetic energy is maximum when roller coaster reaches the bottom most point

Image credits: anonymous, Wooden roller coaster txgiCC BY-SA 3.0

Are rotational and translational kinetic energy equal?

The rotational kinetic energy in most cases has a different magnitude than translational kinetic energy. Although they can be same in some cases.

When the radius of gyration is equal to the radius of rotating object (preferably ring), then we can say that the rotational kinetic energy and translational kinetic energy are equal. Practical examples of this are wheels of bike and cycle while coming down a slope.

Is rotational kinetic energy less than translational energy?

Yes, the rotational kinetic energy is always lesser than or equal to the translational kinetic energy. This is because the ratio of radius of gyration to the radius of rolling body maximizes at 1 which is for a ring or hoop.

We have discussed the formulae of rotational kinetic energy and translational kinetic energy in the above sections of this article We can find their ratio by dividing both the formulae with each other. Let us discuss more about the ratio in further sections of this article.

What is the ratio of the translational kinetic energy to the rotational kinetic energy when it reaches the bottom of the ramp?

When the object reaches the bottom of the ramp, the surface starts becoming horizontal. At this position the value of ratio of rotational kinetic energy and translational or linear kinetic energy will be as per the information given in the section below-

After dividing the formulae of translational kinetic energy and rotational kinetic energy, we get the following ratio- 2/5. This is the required ratio that is ratio of translational kinetic energy to rotational kinetic energy.

How much fraction of the kinetic energy of rolling body is purely translational and rotational?

If the body is following a rolling motion without any slipping and the surface on which it is moving is horizontal then the ratio of linear kinetic energy to rotational kinetic energy is 2:1.

The required value is calculated by dividing the two formulae discussed above, one of translational kinetic energy and other of rotational kinetic energy. Hence we can say that the translational kinetic energy is twice in magnitude of rotational kinetic energy.

What fraction of the total kinetic energy of a rolling sphere is translational

Let us assume that the sphere that is rolling is solid. The moment of inertia for a solid sphere is given by 2/5 mr^2. After finding the rspective values of translational kinetic energy and rotational energy we add them.

To find the value of total kinetic energy we can simply add the values of both the kinetic energies. When we divide the values of total kinetic energy of this rolling sphere and its translational kinetic energy, we get the ratio as- 5/7.

Can an object have rotational and translational kinetic energy

Yes. We can see multiple examples of the same in our everyday lives. A moving object can have both the types of kinetic energies that is rotational and kinetic. 

The most common example of the above case is a wheel or sphere rolling. When an object is following a rolling motion without slipping, it is possible for it to have both the types motion that is rotational and translational. Most common example regarding this is wheel. A wheel rotates as well as moves in a straight line motion. Implyng that it will have both the kinds of kinetic energies- rotational one and translational one. 

The ratio of rotational and translational kinetic energy of a rolling circular disc

The moment of inertia for a circular disc is given as – Mr^2/4.  We can obtain the value of rotational kinetic energy after substituting the value of moment of inertia in the formula discusse above.

Once we are done with dividing the values of rotational kinetic energy and translational kinetic energy, we will get our desired value.The ratio we get after dividing both these terms is 1:2. This has been already discussed in the above sections of this article.

What would be the ratio of the rotational kinetic energy and translational kinetic energy of the rolling solid cylinder

The moment of inertia of a solid cyinder is 1/2 mr^2. After putting the value of moment of inertia we can easily find out the value of rotational kinetic energy. 

After dividing the values of rotational kinetic energy and translational kinetic energy we get the required answer. The answer we get is 1:2. That means the translational kinetic energy is twice the value of rotational kinetic energy of the solid cylinder.

Abhishek

I, Abhishek Khambhata, have pursued Btech in Mechanical Engineering. Throughout four years of my engineering, I have designed and flown unmanned aerial vehicles. My forte is fluid mechanics and thermal engineering. My fourth year project was based on performance enhancement of unmanned aerial vehicles using solar technology. I would like to connect with like minded people.

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