Hoop Loop the Loop

Classical Mechanics Level 3

A hoop H of radius r rolls without slipping down the incline. The starting height h (measured from the bottom of the hoop) is such that the hoop acquires a velocity just sufficient to "loop the loop" - i.e., the hoop just maintains contact with the circular track of diameter d at the top. What is h?

d − r d-r 3 2 d − 3 r \frac{3}{2}d-3r 3 2 d − r \frac{3}{2}d-r d d

Alex Wang by Alex Wang , 25, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Alex Wang
Nov 21, 2014

The thing that's tricky about this problem is that the hoop has a radius r.

Writing Conservation of Energy, m g ( h + r ) = m g ( d − r ) + 1 2 m v 2 + 1 2 m r 2 ω 2 mg(h+r)=mg(d-r) + \frac{1}{2} m v^2 + \frac{1}{2} m r^2 \omega^2 .

In order to maintain contact at the top of the loop, the centripetal force must balance the force of gravity. m v 2 d 2 − r = m g . m \frac{v^2}{\frac{d}{2}-r} = mg.

We also use the fact that ω r = v \omega r = v .

Using these two equations and solving for h, we get h = 3 2 d − 3 r h=\boxed{\frac{3}{2} d - 3r} .

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...