# Moment of Inertia of a Rotating Cam

• Dorothy Weglend
In summary: We first calculate the moment of inertia around the center of the disk (Ic), then add the moment of inertia of a disk with a hole in it (Imh). Because the hole is located at a distance r from the center of the large disk, the total moment of inertia is (Imh+r*Ic).
Dorothy Weglend
This cam is a circular disk rotating on a shaft that does not pass through the center of the disk. It is manufactured by first making the cam with radius R, then drilling an off-center hole, radius R/2, parallel to the axis of the cylinder and centered at a point R/2 from the center of the cam.

The cam, of mass M is then slipped onto the circular shaft and welded into place. What is the kinetic energy of the cam when it is rotating with angular speed w about the axis of the shaft?

This is giving me real problems, so I would appreciate any suggestions.

I used Icm = MR^2/2 for the moment of inertia of a solid, rotating cylinder around its center of mass, and the parallel axis theorem to get:

I = MR^2/2 + MD^2 = MR^2/2 + M(R/2)^2 = 3MR^2/4

From which the rotational kinetic energy should just be:

Kw = (3MR^2/4)w^2(1/2) = (3MR^2/8)w^2

But the answer in the book is (23/48)MR^2w^2

So, wow, am I off.

The only thing I can think of is that the shaft might be made of a different material, so I would have to change the mass calculation somehow, but the problem doesn't say anything about that. Anyway, wouldn't they make these things out of the same stuff?

Thanks for any help,
Dorothy

I think they want the KE of the cam, not the cam plus shaft.

A hint and a trick. Hint: M is the mass of the cam, not of a solid disk of radius R. Trick: A disk with a hole can be thought of as a solid disk of radius R plus another solid disk of radius R/2 but with negative mass.

Well, that was easy. Thank you! Cool trick, that negative mass idea.

Does this actually have any kind of physical significance? Wouldn't the effective KE of the cam be different, because really, it seems to me, the shaft and the cam form a single object. Is there any point to this calculation besides the routine torture of physics students?

Thanks again, Doc Al. I hope you had a great thanksgiving.

Dorothy

Request for full solution

Can you show me the full solution of this question?

Thanks :-)

So just to follow up on what Doc Al posted. When he is talking about the disk of radius R being subtracted with the missing disk we are assuming that the original disk does not follow the inertia of a regular disk right? Since the axis it is being turned on is not at the center of the large disk we use MR^2 instead of 1/2MR^2 right? Just double checking.

Thanks Matt

matt0101 said:
When he is talking about the disk of radius R being subtracted with the missing disk we are assuming that the original disk does not follow the inertia of a regular disk right?
A disk with a hole in it has a different rotational inertia than a complete disk, if that's what you are asking.
Since the axis it is being turned on is not at the center of the large disk we use MR^2 instead of 1/2MR^2 right? Just double checking.
To find the rotational inertia of a disk about a point not at its center, we use the parallel axis theorem.

## What is moment of inertia and why is it important in a rotating cam?

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. In a rotating cam, it is important because it determines the amount of torque needed to accelerate or decelerate the cam.

## How is moment of inertia calculated for a rotating cam?

The moment of inertia of a rotating cam can be calculated by multiplying the mass of the cam by the square of its distance from the axis of rotation. It can also be calculated by integrating the mass distribution of the cam over its entire volume.

## How does the shape of a rotating cam affect its moment of inertia?

The shape of a rotating cam can greatly affect its moment of inertia. A larger radius or more mass distributed further from the axis of rotation will result in a higher moment of inertia, while a smaller radius or more mass concentrated closer to the axis of rotation will result in a lower moment of inertia.

## How does the moment of inertia of a rotating cam affect its performance?

The moment of inertia of a rotating cam can greatly affect its performance. A higher moment of inertia will require more torque to accelerate or decelerate the cam, which can affect the speed and efficiency of the cam's movement. A lower moment of inertia can result in smoother and faster movement.

## How can the moment of inertia of a rotating cam be optimized for a specific application?

The moment of inertia of a rotating cam can be optimized for a specific application by carefully choosing the shape, size, and distribution of mass in the cam. Reducing the mass and/or moving it closer to the axis of rotation can result in a lower moment of inertia, while increasing the mass and/or moving it further from the axis of rotation can result in a higher moment of inertia.

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