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

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SUMMARY

The discussion focuses on calculating the height from which a solid brass ball must be released to successfully complete a loop-the-loop track. The ball has a mass of 0.280 g and the loop has a radius of 14.0 cm. The key equations used include the forces at the top of the loop and the conservation of energy principles. The initial calculations yielded a height of 0.350 m, but the correct answer is 37.8 cm, which requires accounting for rotational inertia in the kinetic energy calculation.

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labgoggles
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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, FN + mg = ma
Fc = mv2 / R where R is radius of the track

Ui + Ki = Uf + Kf

The Attempt at a Solution


First, using FN + 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 = FN.

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 Ui + Ki = Uf + Kf, 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!
 
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labgoggles said:
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.
 
Thank you so much! I was able to work it out.
 

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