- #1

JessicaHelena

- 188

- 3

## Homework Statement

Please look at the problem attached as a screenshot.

## Homework Equations

Assuming frictionless, Ei = Ef, which means objects that are the same will end up in the same heights (so we can group A&C, B&D, and E&F).

For A&C and E&F, mgh = KE_rot + KE_trans

For B&D, it is mgh = KE_trans.

Also, v = rw (I know it's omega, but for convenience, I'll write it as w).

I _solid sphere = 2/5mr^2

I_hollow sphere = 2/3mr^2

## The Attempt at a Solution

After all these equations set up, now it's pretty much plugging things in. For A&C ,

mgh = 1/2mv^2 + 1/2Iw^2

mgh = 1/2mv^2 + 1/5mr^2w^2

gh = 1/2v^2 + 1/5 r^2(v/r)^2

gh = 1/2v^2 + 1/5v^2 = 7/10v^2

so v^2 = 10gh/7

then KE at the end is then 1/2mv^2 = 5mgh/7, and that can be converted to the new GPE. so the height will be 5/7 the original height.

For E&F, I can use a similar process, only now I = 2/3mr^2. That gives me a KE of 3/5mgh, so the new (final) height will be 3/5 the original height.

For the blocks (B&D), there's only KE_trans, so Ei = Ef and mgh = 1/2mv^2. With no rot KE to lose trans KE to, the final height should be the same as the original height.

Thus, in order, it is B&D, A&C, E&F.

Could someone please check if I don't have any flaws in my reasoning?