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I have a question about comparing different objects in rotational motion. In this scenario:

A cylinder (with moment of inertia = $\frac{1}{2}MR^2$), a sphere ($\frac{2}{5} MR^2$) and a hoop ($MR^2$) roll down the same incline without slipping. All three objects have the same radius and total mass.

*1) At the bottom of the incline, which object has the greatest translational kinetic energy?

2) Which object experiences the greatest rotational acceleration?

3) Which object needs the greatest coefficient of static friction to prevent it from slipping?*

For (1), I feel like the sphere should have the greatest translational energy and the least rotational energy because it has the greatest moment of inertia. However, according to 2), the objects have different angular acceleration and speeds at the bottom of the ramp—which makes me unsure of 1) since rotational KE = $I \cdot \omega^2$. So I don't know if it's possible to compare them using just the moment of inertia.

3) is the most difficult for me.

Is there any way to show the answers to these questions mathematically?

Ganesh
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matchstick
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1 Answers1

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Use the torque equation about a inertial frame of reference, in this case the point of contact. You can see that two forces will act, as I've shown. From the picture you can see that angular acceleration depends inversely on the moment of Inertia, hence in this case the sphere will have the maximum angular acceleration,hence the maximum energy (rotational),and this rotational energy will culminate as the translational energy too.enter image description here

Abhinav
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