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## Homework Statement

A block of mass m slides down a hemisphere of mass M. What are the accelerations of each mass? Assume friction is negligible.

(See attatchment)

## Homework Equations

[itex] a_M [/itex] = Acceleration of hemisphere

[itex]N_m [/itex] = Normal force of M onto m

[itex]N_M [/itex] = Normal force of ground onto M

[itex]\sum \text{F}_x = ma[/itex]

## The Attempt at a Solution

So from the FBD's, I come up with

[itex]\sum \text{F}_{xm}= mg\sin \theta = m(a_t - a_M \cos \theta)[/itex]

[itex]\sum \text{F}_{ym} = N_m - mg \cos \theta = -m(a_r + a_M \sin \theta)[/itex]

[itex]\sum \text{F}_{xM} = -N_m \sin \theta = Ma_M[/itex]

I need another equation, so I tried going the route of work-energy, to find the tangential speed of the block sliding on the hemisphere, but can I look at the energy of the block by itself? I figure I cannot, as it is atop an accelerating body.

If I can consider the energy of the block by itself to find the tangential speed, then I can solve for aM,

$$ a_M = gm\sin \theta \frac{2-3\cos \theta}{M-m\sin ^2 \theta} $$

which goes to 0 when M >> m and so then [itex]a_t = g\sin \theta[/itex] in that case which checks out, however I am still a little weary about this.

I'm rather stuck here so any help would be appreciated.

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