How High Must the Brass Ball Start to Complete the Loop-the-Loop?

[labgoggles]

Hi everyone, I've been working on this problem for a while now, and I was hoping someone here could point me in the right direction. Here goes:

1. Homework Statement
A solid brass ball of mass 0.280 g will roll smoothly along a loop-the-loop track when released from rest along the straight section. The circular loop has radius R= 14.0, and the ball has radius r <<R. (a) What is h if the ball is on the verge of leaving the track when it reaches the top of the loop?

I don't have units of R, but I think it's in cm.

Homework Equations

At the top of the loop, F_N + mg = ma
F_c = mv² / R where R is radius of the track

U_i + K_i = U_f + K_f

The Attempt at a Solution

First, using F_N + mg = ma, I know ma is (mv²)/R, and (mv²)/R = mg if the ball is going to fall at the top, from (mv²)/R - mg = F_N.

I plug in R = .14m, m= .280 X 10^-3g, and g to get v² = 1.37 m/s. I think it's correct up to here...

However, in U_i + K_i = U_f + K_f, initial U = mgh, final U =mg(2R) because the ball is at the top of the loop, initial K = 0, but final K = 1/2mv². When I plug everything in and m cancels out,

gh = 1/2v² + g(2R)

(9.8m/s²) = 1/2(1.37m²/s²) + (9.8m/s²)(2 X .14 m)

and h = .350 m or 35.0 cm. My answer is close to the book's, which gives h = 37.8 cm, but I think I am missing something. Do I need to include rotational inertia, (1/2)Iω² as part of K final? If I did, it looks like I would need to find I and omega, and I need the radius of the ball to do that (not given). I only have its mass and calculated velocity. Please let me know if I need to give more information and clarify, thank you!

 

[haruspex]

Do I need to include rotational inertia, (1/2)Iω² as part of K final? If I did, it looks like I would need to find I and omega, and I need the radius of the ball to do that (not given). I only have its mass and calculated velocity.

Yes, that's the error.
You don't need to know r. Put it in as an unknown and it will cancel out. For a rolling uniform ball, there is a constant fraction of its KE that is in the rotation. Relate omega to linear speed in the usual way.

 

[labgoggles]

Thank you so much! I was able to work it out.