What are the Moments of Inertia for a Ball and Hoop on a Ramp?

[vector7]

Homework Statement

A hollow, 50N ball and a 50N hoop are released from rest at the top of a 3m long ramp. Both objects have a diameter of .5m. If the ball reaches the bottom in 5s, and the hoop in 7s, find (for each object):
a. velocity at the bottom
b. angular speed at bottom
c. moment of intertia at bottom
d. rotational kinetic energy at bottom
e. angular momentum at bottom

Homework Equations

d=1/2at^2+volt
I=mr^2

That's another part of the problem: I don't know the needed equations

The Attempt at a Solution

I answered part A correctly, using d=1/2at^2, but the rest I bombed. I attempted using 2πr/t for part B, I=mr^2 for C, KE=1/2mv^2 for D, and mvr for E, all of which were wrong.
Any help would be appreciated.

 

[Stovebolt]

Homework Statement

A hollow, 50N ball and a 50N hoop are released from rest at the top of a 3m long ramp. Both objects have a diameter of .5m. If the ball reaches the bottom in 5s, and the hoop in 7s, find (for each object):
a. velocity at the bottom
b. angular speed at bottom
c. moment of intertia at bottom
d. rotational kinetic energy at bottom
e. angular momentum at bottom

Homework Equations

d=1/2at^2+volt
I=mr^2

That's another part of the problem: I don't know the needed equations

The Attempt at a Solution

I answered part A correctly, using d=1/2at^2, but the rest I bombed. I attempted using 2πr/t for part B, I=mr^2 for C, KE=1/2mv^2 for D, and mvr for E, all of which were wrong.
Any help would be appreciated.

From the sound of things, I assume there should have been a stipulation that there is no slipping occurring (that is, all motion is a result of rolling only, not sliding and rolling). Otherwise, portions of the problem would be indeterminate without additional information.

That said, for part B, your formula 2πr/t was not quite correct, but your thoughts are probably on the right track. The units of that formula would be m/s, which is linear velocity, but what you are looking for is angular velocity, which would be in units of radians/s. Think about the relationship between the linear velocity at the bottom and how fast the objects must be rotating to maintain that speed.

For Part C, are you expected to actually calculate the moments of inertia? If so, things will be a bit more complicated. If not, your formula should be correct for the hoop (assuming a thin-walled hoop, if it has a given thickness you will need to take the average radius). You should be able to look up the moment of inertia for a hollow sphere.

Parts D and E will require the moments of inertia for each object from Part C. You should have formulas for both Energy and Momentum. Remember, they are similar in nature to linear Energy/Momentum equations ( E = (1/2) m v² and p = m v).

I hope that gets you on the right track. If I've confused you, please let me know.

 

[vector7]

Alright I mostly understand everything now, thanks. But I still have a problem on part C, with the moment of inertia at the bottom. For the ball, i used I=2/3mr^2, which yielded 2.08, and for the hoop I used I=mr^2 and got 3.13, but I was still marked wrong. I'm not sure wether the work is wrong or wether it was marked just because I left out the units.