- #1

Dustgil

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## Homework Statement

A solid homogeneous hemisphere of radius a rests on top of a rough hemispherical cap of radius b, the curved faces being in contact. Show that the equilibrium is stable if a is less than 3b/5.

## Homework Equations

V = mgh

## The Attempt at a Solution

So the center of mass of a solid hemisphere is 3a/8. This hemisphere is upside down and resting on top of another hemisphere of radius b. Therefore if we define the zero point of the potential energy to be level with the bottom of the lower hemisphere, the height at equilibrium of the center of mass is h = 5a/8 + b.

As hemisphere a rotates, the center of mass is displaced an angle theta to the initial point of contact. theta is our generalized coordinate. This is where things get shaky. the line from the center of mass to the original point of contact and the line normal to the surface form a right triangle with angle theta. the normal line is part of the line contributing to the height of the center of mass. It seems like this could be equal to

[tex] \frac {5a} {8} * cos\theta[/tex]

but I'm not so sure. maybe its the way I've drawn the picture, but it seems like a chunk is still missing out of the line if I go with that.

For the other contribution, since the point of contact is at a different location on hemisphere b, this part cannot just be equal to b. I make the supposition (that I feel unsure of) that the distance from point of contact the center of curvature makes the same angle theta with the normal. I reason this because they are both spherical in shape and their center of curvatures are located at their centers of mass. But I feel like this only holds true when the spheres have the same radius..so I'm not sure on that. At any rate, I get

[tex] bcos\theta[/tex]

for the height contribution at this point. Taken together, the potential energy function is

[tex]V = mgcos(\theta)(\frac {5a} {8} + b)[/tex]

which agrees with my original thoughts on when theta = 0 but doesn't give me the correct answer when I determine the minimum of the function. So, I messed up in some spots (probably at least the two I mentioned). Any help?

As an addendum: If anyone has any helpful resources for solving advanced trigonometry problems I'm all eyes. That'd be much appreciated.