Which Object Reaches the Bottom of an Incline Fastest?

[mpm]

I have a mechanics question that I can't seem to figure out. I've spent quite a bit of time on it but don't have much of an answer.

If anyone can help I would appreciate it.

Round objects are rolling without slipping down an inclined plane of height H above the horizontal. The box is sliding without friction down the slope. All round objects have the same radius R & the same M, which is also the mass of the box. The moments of intertia for the round objects are: Hoop: I = MR^2, Cylinder = I (1/2)MR^2, Sphere I = (2MR^2)/5. The 4 objects are released, one at a time, from the hiehg H. Which one arives at the bottom with the greatest speed? Why? Which arrives with the smallest speed? Why? What physical principle did you use to answer these questions?

The professor wants answers in words and not so much equations.

I would think the box would be the slowest. If I remember right, an object rotates faster if the mass is in the center instead of on the outside edges.

But I am really confused.

If anyone can help I would appreciate it.

Thanks,

Mike

 

[jamesrc]

Have you covered conservation of energy yet?

Assuming you have, you know that all of the objects start with the same potential energy (MgH). All of this potential energy is converted into kinetic energy by the time it reaches the bottom (no frictional losses). This kinetic energy can be thought of as the sum of translational kinetic energy (mv²/2) and rotational kinetic energy (Iω²/2). From kinematics, you know that the rotational speed is related to the translational speed (ω=v/R) for no slip.

You are asked about the relative speeds of the objects, so you are interested in v, the tranlsational speed. If you look at the conservation of energy, you will find out how the magnitude of the object's moment of inertia affects the relative contribution of rotational to translational kinetic energy and, therefore, the final value of the speed (you don't have to actually solve it, you just need to see the relationship). I hope that helps.

Hint: the block is not the slowest; it does not rotate (has no rotational kinetic energy).