Question

A solid rod and a hollow rod (pipe) are released from rest at the same time at the same distance from the bottom of the slope. Which will win the race?

a. The hollow rod wins every time
b. The solid rod wins every time
c. The solid rod wins if it has more mass than the hollow rod it is racing
d. The hollow rod wins if it has more mass than the solid rod it is racing
I know it has something to do with the moment of inertia equations, but i cant find it for hollow rod.

Answer 1

Do you know the moments of inertia for a solid disk and a ring? They would be analogous.

Answer 2

no :(

Answer 3

im sure i can look them up tho :)

Answer 4

But i do have I=mr^2 for hollow cylinder and I=1/2mr^2 for a solid cylinder

Answer 5

So the hollow would have a greater moment of inertia, does that mean that it is always slower?

Answer 6

Yep. Do you understand why?

Answer 7

Yeah i think so because the moment of inertia, is resistance to speed. Also does the mass matter?

Answer 8

Well, let's look at what happens. We'll assume no slipping.
The potential energy (mgh) goes into the translational kinetic energy, as well as the rotational kinetic energy.

\[mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2\]
and
\[\omega = \frac{v}{r}\]

If you expand the first equation using the moment of inertia (I) for each shape, you can solve for v to find the translational velocity at the bottom. The faster one wins!

Answer 9

ok so mass cancels out?

Answer 10

yep!

Answer 11

awesome thanks SO much! I understand it now!!!!

Answer 12

Excellent! It was my pleasure