Sliding on a sliding hemisphere

[Nanyang]

Homework Statement

A small mass m slides on a hemisphere of mass M and radius R is also free to slide horizontally on a frictionless table. An imaginary vertical line is drawn from the center of the hemisphere to its highest point, where the small mass is originally placed at rest. The angle A is the acute angle made by another imaginary line drawn from the small mass after it starts to slide down to the center of the hemisphere with the imaginary vertical line mentioned before. In other words, A= 0 originally. Given that cosA = k. Find the ratio of M/m.

Homework Equations

I'm not really sure.

The Attempt at a Solution

Here's what I did.

I imagine that the big mass moves at velocity V and the small mass with a tangential speed v after some time. Then using the conservation of energy I obtain,

mgR = $$\frac{1}{2}$$(M+m)V² + $$\frac{1}{2}$$mv²EDIT: I just spot my mistake on the mgR thing.

Next I obtain the condition when the small mass just begins to leave the hemisphere's surface,

gcosA= v²/R

I think my mistake is the above.

I then obtain another equation using the conservation of momentum in the left-right direction,

mvcosA=(M+m)V

Then I substituted and rearranged all the stuff and got:

$$\frac{(k-1)}{k(2-k)}$$² = M/m

But it doesn't look correct.

EDIT: I think I found the mistake in the conservation of energy part. But still do give hints on how to solve it.

 

[Nanyang]

I just solved it. Anyway, typing the problem down again here sure does help in solving it by making you think hard again. :rofl:

 

[tomcue]

I would approach this problem by first identifying the relevant physical principles and equations that can be applied. In this case, the key principles are conservation of energy and conservation of momentum.

Using conservation of energy, we can write:

mgh = (1/2)(M+m)V^2 + (1/2)mv^2

where h is the height of the small mass above the center of the hemisphere, V is the velocity of the center of the hemisphere, and v is the tangential velocity of the small mass.

Next, we can use conservation of momentum in the horizontal direction:

mv = (M+m)V

where v is the horizontal velocity of the small mass and V is the velocity of the center of the hemisphere.

Combining these two equations, we can eliminate V and solve for the ratio M/m:

M/m = (k-1)/(k^2)

where k = cosA.

This result makes intuitive sense - as the angle A (and therefore k) decreases, the ratio M/m also decreases, indicating that a larger mass M is required to keep the smaller mass m in equilibrium.

In conclusion, as a scientist, I would approach this problem by carefully considering the relevant principles and equations, and using them to derive a solution that is consistent with our understanding of the physical world.