MULTIPLE CHOICE: Rotational Inertia

[rvhockey]

1. Which rolls to the bottom of a hill sooner, a car tire alone or the same tire mounted on a rim?
a. The mounted tire, provided the tire is more massive than the rim
b. The tire alone, provided it is as heavy as the rim
c. The mounted tire, provided the tire is as heavy as the rim
d. The tire alone, regardless of its weight
e. The mounted tire, regardless of its weight

2. A ring and a disk roll down a hill together. Which reaches the bottom first?
a. Both reach the bottom at the same time
b. Depends on the moments of inertia
c. The disk
d. Depends on the masses
e. The ring

3. A ring, a disk, and a solid ball having equal masses roll down a hill at the same time. Which reaches the bottom first?
a. Depends on what each is made of
b. The disk
c. The ring
d. Depends on the radius of each
e. The ball



The only ones I'm using are
I_ring = MR²
I_cyclinder = (1/2)MR²
I_sphere = (2/5)MR²



For number 1, my teacher said it wasn't d, so I have it down to e, because a cyclinder (mounted tire) has a smaller rotational inertia.
For number 2, my teacher said it wasn't b, so I'm thinking its c because of the cyclinder again.
For number 3, my teacher said it wasn't d, and I have no clue what else it could be, because if the radii are different enough, can't all the inertias be equal?

Any thoughts would be greatly appreciated

 

[Mthees08]

Hint3: Radii often cancel out. If we drew a free body diagram we possibly would see that this problem is entirely dependent on M.

The first two I have no real hint on that won't give it away, but think again more about mass, and less about radii.

 

[heth]

2. A ring and a disk roll down a hill together. Which reaches the bottom first?
a. Both reach the bottom at the same time
b. Depends on the moments of inertia
c. The disk
d. Depends on the masses
e. The ring

3. A ring, a disk, and a solid ball having equal masses roll down a hill at the same time. Which reaches the bottom first?
a. Depends on what each is made of
b. The disk
c. The ring
d. Depends on the radius of each
e. The ball[/b]

You could make yourself a ramp and try this? OK so you couldn't do that in an exam, but it'll give you a better feel for what's going on, and improve your intuition for similar problems in the future.

 

[hty21]

You could make yourself a ramp and try this?

I'm not sure why this is true, but when the teacher demonstrated rolling objects down a ramp in class a few days ago, we saw that different objects with the same shape/material but different radii (we used black rubber cylinders of different sizes) reached the bottom of the ramp at the same time. The moments of inertia (0.5 M R^2) are different, so why would they travel at the same rate? How does the moment of inertia affect the speed of the object (linear/rotational)?

Thanks!

 

[Mthees08]

I'm not sure why this is true, but when the teacher demonstrated rolling objects down a ramp in class a few days ago, we saw that different objects with the same shape/material but different radii (we used black rubber cylinders of different sizes) reached the bottom of the ramp at the same time. The moments of inertia (0.5 M R^2) are different, so why would they travel at the same rate? How does the moment of inertia affect the speed of the object (linear/rotational)?

Thanks!

I am pretty sure I alluded to that already. but in reality it affects the rotational acceleration, which when going down an inclined plane is key. Assuming there is no slippage. We know torque=I(alpha) I am sorry i do not know how to insert the proper symbols. So a lower I gives you a higher Alpha under the same torque. The question for you is how is the torque different if it is at all in these scenarios, what determines the torque on the objects, what forces are acting on them. So therefore again I say what I always say, draw a free body diagram, find torque.