A particle of mass m rests atop a frictionless hemisphere of radius R. A force F = -mkyα acts in the y direction. After an initial small displacement, the particle slides down the sphere under the action of the force.
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.
I would assume relevant equations include a summation of forces:
ΣF = FN - mgcosθ - mkyαcosθ
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 - mkyαcosθ = mac
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 + 1/2mv2
mgR = mgy + 1/2mv2
I figured y= Rcosθ and simplified the energy equation to:
v2 = 2gR(1-cosθ)
Which is helpful when I know that ac = v2/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.