# Determining Minimum Height for Marble to Complete Loop-the-Loop

• kamikaze1
In summary, to find the minimum height h for a marble of mass m and radius r to make it around a loop-the-loop of radius R, we must take into account the center of mass and use the constraint of velocity at the top of the loop. The final energy at the top can be calculated using the kinetic energy and rotational energy. Setting this equal to the initial energy, we can solve for h.

## Homework Statement

A marble rolls down a track and around a loop-the-loop of radius R. The marble has mass m and radius r. What minimum height h must the track have for the marble to make it around the loop-the-loop without falling off?

## Homework Equations

What minimum height h must the track have for the marble to make it around the loop-the-loop without falling off?

## The Attempt at a Solution

Find initial energy: mgh
Final energy tabulation: mg(2R) + 0.5mrg + 0.5(2/5)mr^(2)(v/r)^2
Set equal to initial energy, I obtain 2R+7/10 =h, which isn't the right answer, what did I do wrong?

I found the velocity at top by using Newton's 2nd law, i get v = squareroot(rg). I find w using constraint of wr=v. I get 2/5(mr^2) using moment of inertia for a solid sphere.

Any help would be appreciated. I'm so confused why this doesn't work out.

I'm still new to this, but it looks like you've got everything right except for the last part of your final energy calculation (the 0.5(2/5)mr^(2)(v/r)^2 part). I'm not quite sure why it's there, but I'm pretty sure that's the part that's throwing you off.

SchruteBucks said:
I'm still new to this, but it looks like you've got everything right except for the last part of your final energy calculation (the 0.5(2/5)mr^(2)(v/r)^2 part). I'm not quite sure why it's there, but I'm pretty sure that's the part that's throwing you off.

The last part is the kinetic rotational energy using I = (2/5)(M)(R^2)
Using kinetic rotational energy = 0.5Iw^2, I have
0.5(2/5)(MR^2)w.
using constraint wr=v, knowing that v=(rg)^0.5, I get
0.5(2/5)mr^(2)(v/r)^2

The energy at the top of the loop is the potential (which you have) plus it's kinetic, since it is moving in a circle there must be an acceleration at the top equal to $\frac{V^2}{R}$, for this to be a minimum (IE only just making it around the loop) this acceleration must be set equal to $g$ so we have
$$V^2 = gR \Rightarrow KE = \frac{1}{2}mgR$$
$$\displaystyle{\dot{. .}} mgh = mg(2R) + \frac{1}{2}mgR$$
Which you can then solve for the minimum height :)

EDIT: Just reread your question, the "effective" radius of the loop is changed because of the radius of the marble (the center of mass of the marble moves around a slightly smaller loops) and there is indeed rotational energy equal to $\frac{1}{2}\frac{2}{5}\frac{gR}{r^2}mr^2$ (again this has to be changed because the $R$ isn't really the radius of the loop that the center of mass moves over, but I'll leave that to you)

:)

Last edited:
Never mind. Thanks for all your suggestions. I've just solved them. I left out the little tiny details which messed up the problems. I didn't take into account of the center of mass, which I paid dearly.
Here is how I solved it.
Initial energy: mg(h+r)
At the loop, the radius is R-r because we're taking into account of the center of mass. Hence using Newton's 2nd law.
m(v^2)/(R-r) = mg, solve for v = (g(R-r))^0.5 (1)

Final energy at top of the loop is:
mg(2R-r) + 0.5(m)(g(R-r)) + 0.5(2/5)(mr^2)(w^2).
Using velocity constraint, w=v/r=(1/r)(g(R-r))^0.5
Plug that in (1), and set (1) equal to initial energy, then solve for h.