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## Homework Statement

A particle of mass

*m*rests atop a frictionless hemisphere of radius

*R.*A force

*F = -mky*acts in the y direction. After an initial small displacement, the particle slides down the sphere under the action of the force.

^{α}Find a) the height above the equator and b) the speed of the particle, both at the moment that it leaves the surface of the hemisphere.

## Homework Equations

I would assume relevant equations include a summation of forces:

ΣF = F

_{N}- mgcosθ - mky

^{α}cosθ

## The Attempt at a Solution

I feel that the key is that the normal force is equal to 0 when the particle leaves the surface of the hemisphere as there is no longer contact between the two. Thus you get:

ΣF = - mgcosθ - mky

^{α}cosθ = ma

_{c}

at the point when the particle leaves the surface of the hemisphere.

I would also assume that energy is a part of the equation:

E(y=R) = mgR

E(y) = mgy + 1/2mv

^{2}

mgR = mgy + 1/2mv

^{2}

I figured y= Rcosθ and simplified the energy equation to:

v

^{2}= 2gR(1-cosθ)

Which is helpful when I know that a

_{c}= v

^{2}/R

Now I assume somewhere in that logic I went wrong because when I try using the energy equation and force equation to set up something where I might find cosθ but it keeps coming out to be a very, very ugly equation which I don't see as plausible. My professor hinted that I could also set the problem up in the non-inertial frame of the particle as it slides down but I wouldn't be sure where to start there. Any help is hugely appreciated.