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).

 

[quicknote]

Based on the given information, the object with the greatest speed at the bottom of the inclined plane would be the sphere. This is because the sphere has the smallest moment of inertia out of the four objects, meaning it has less resistance to rotation and can roll down the incline faster than the other objects. This is due to the fact that the majority of its mass is concentrated at the center, which allows it to rotate more easily compared to the hoop and cylinder which have more mass distributed away from the center. The box, being a sliding object, would have the smallest speed as it does not have a rotational component and is only affected by the force of gravity.

The physical principle used to determine the speeds of the objects is the conservation of energy. As the objects roll down the incline, their potential energy is converted into kinetic energy. The object with the smallest moment of inertia (the sphere) will have the greatest speed at the bottom due to its ability to efficiently convert potential energy into rotational kinetic energy. This principle is also used to explain why the box, with no rotational component, will have the smallest speed at the bottom.

It is important to note that this answer is based on the assumption that the objects are all released from the same height and there is no slipping or friction involved. If these factors were to be taken into account, the results may differ.