Kinetic Energy rotational and translational conceptual question.

[bdh2991]

Homework Statement

A ball rolls without slipping down incline A, starting from
rest. At the same time, a box starts from rest and slides down incline B, which
is identical to incline A except that it is frictionless. Which arrives at the bottom
first? (a) The ball arrives first. (b) The box arrives first. (c) Both arrive at
the same time. (d) It is impossible to determine.

Homework Equations

KE = 1/2 I ω²

KE = 1/2 m v²

The Attempt at a Solution

I answered D because i felt that you would need to know how much work had been done by friction. The correct answer is B however, and I'm not certain why. Does the ball rolling down the incline have both rotational and translational energy?

 

[freddyfish]

Yes, the ball has got both translational and rotational energy. It is the extra term (the rotational energy) that makes the ball reach the "bottom" second. This is a result of the first law of thermodynamics, which states that energy cannot be created or destroyed. :)

 

[bdh2991]

Yes, the ball has got both translational and rotational energy. It is the extra term (the rotational energy) that makes the ball reach the "bottom" second. This is a result of the first law of thermodynamics, which states that energy cannot be created or destroyed. :)

But wouldn't that mean it has more kinetic energy therefore it would be moving faster?

 

[freddyfish]

No, the rotational energy and the translational energy must share the amount of energy equal to the decrease in potential energy.

 

[freddyfish]

For the ball:

potential energy --> kinetic energy + rotational energy (both are nonnegative quantities).

Thus, the translational energy would be greater in the absence of the rotational energy.For the box, however: potential energy --> kinetic energy + rotational energy (where the rotational energy evidently is zero.) Thus, "maximal" kinetic energy.

Since the mass cancels out in both cases, the latter case involves a translational energy that is greater per unit mass and thus the box travels at a higher speed.

 

[bdh2991]

No, the rotational energy and the translational energy must share the amount of energy equal to the decrease in potential energy.

ok so basically the rotational + translation in the rolling ball would equal the translational of the sliding block, however because the rolling ball is losing energy to friction the box would be faster?

 

[haruspex]

ok so basically the rotational + translation in the rolling ball would equal the translational of the sliding block, however because the rolling ball is losing energy to friction the box would be faster?

Well, no, it's not losing energy to friction exactly. The friction is causing some the energy to go into rotational KE instead of linear KE. So the linear acceleration is less than if the ball were also on a frictionless surface. If the slopes were to level out then rise again, the two objects would reach the same height, but the ball would take longer to get there.