# The moment of inertia of a rod that is rotating off the end of the rod.

• Lewis Edmunds
In summary, to find the moment of inertia of an ice skater's arms while spinning, you can model them as two rod-like arms attached to a cylinder representing the torso. If you know the moment of inertia of a rod about its center of mass, you can use the parallel axis theorem to find the moment of inertia about the desired axis. Otherwise, you can directly integrate to find the moment of inertia.
Lewis Edmunds
Is there a formula for the moment of inertia? A thin, uniform density rod is rotating about an axis that is off the end of the rod, so it looks a bit like this:

------- |
(------- is the rod and | is the axis of rotation, so the rod is rotating out of the plane of your screen)

I just have a problem about an ice skater spinning and moving their arms, and I'm not sure how to work out the moment of inertia of the arms. The problem says to model them as two rod-like arms that are attached to the the outside of the torso (modelled as a cylinder)

Do you know the moment of inertia of a rod about its center of mass? If so, use the parallel axis theorem to find the moment of inertia about the axis that you need.

If you don't know the parallel axis theorem, you can find the MMOI directly by integration.

## What is the moment of inertia of a rod that is rotating off the end of the rod?

The moment of inertia of a rod that is rotating off the end of the rod depends on the mass and length of the rod, as well as the distance from the axis of rotation to the end of the rod. It can be calculated using the formula I = (1/3) * m * L^2, where m is the mass of the rod and L is the length of the rod.

## How does the mass of the rod affect the moment of inertia?

The mass of the rod directly affects the moment of inertia, as seen in the formula I = (1/3) * m * L^2. A heavier rod will have a larger moment of inertia compared to a lighter rod with the same length and rotating off the same axis.

## What happens to the moment of inertia if the length of the rod is doubled?

If the length of the rod is doubled, the moment of inertia will increase by a factor of four. This can be seen in the formula I = (1/3) * m * L^2, where increasing the length (L) by a factor of two will result in a four-fold increase in the moment of inertia.

## How does the distance from the axis of rotation to the end of the rod affect the moment of inertia?

The distance from the axis of rotation to the end of the rod is directly proportional to the moment of inertia. This means that the farther the end of the rod is from the axis of rotation, the larger the moment of inertia will be.

## Can the moment of inertia of a rotating rod be changed?

Yes, the moment of inertia of a rotating rod can be changed by altering its mass, length, or the distance from the axis of rotation. For example, by increasing the mass or length of the rod, the moment of inertia will also increase. Similarly, by decreasing the distance from the axis of rotation to the end of the rod, the moment of inertia will decrease.

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