A turntable rotates at a constant angular velocity. Three pennies glued to the surface at three different distances from the center rotate with the turntable. Rank the following according to the moment of inertia of the pennies, with the largest moment of inertia at the top and the smallest at the bottom.

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UpStudy AI · Accepted Answer

To rank the moment of inertia of the pennies glued to the turntable, we need to understand the concept of moment of inertia. The moment of inertia $$ I $$ of a point mass is given by the formula:

$$
I = m r^2
$$

where:
- $$ m $$ is the mass of the object,
- $$ r $$ is the distance from the axis of rotation (in this case, the center of the turntable).

Since the mass of each penny is the same, the moment of inertia will depend solely on the distance $$ r $$ from the center of the turntable. The farther a penny is from the center, the larger its moment of inertia.

### Steps to Rank the Moments of Inertia:

1. Identify the distances $$ r_1, r_2, r_3 $$ of the three pennies from the center of the turntable.
2. Calculate the moment of inertia for each penny using the formula $$ I = m r^2 $$.
3. Rank the pennies based on their distances $$ r $$ from the center.

### Ranking:

- If $$ r_1 < r_2 < r_3 $$, then:
  - Penny at $$ r_3 $$ has the largest moment of inertia.
  - Penny at $$ r_2 $$ has a medium moment of inertia.
  - Penny at $$ r_1 $$ has the smallest moment of inertia.

Thus, the ranking from largest to smallest moment of inertia is:

1. Penny at $$ r_3 $$
2. Penny at $$ r_2 $$
3. Penny at $$ r_1 $$

If you provide the specific distances of the pennies from the center, I can give you a more precise ranking.
The penny farthest from the center has the largest moment of inertia, followed by the one in the middle, and the closest one has the smallest moment of inertia.

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