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A skier starts from rest on a smooth curved ramp that is at a height of R/4 above the base of a smooth fixed hemisphere of radius R. After descending the ramp the skier moves up over the hemisphere. At what angle theta (measured from the vertical through the top of the hemisphere) does the skier leave the surface of the hemisphere?

1. cos⁻¹(2/3)
2. cos⁻¹(5/sqrt(3))
3. cos⁻¹(5/6)
4. cos⁻¹(5/(2*sqrt(3)))

Correct answer: cos⁻¹(5/6)

Solution

Setting centripetal force equal to gravity component and substituting the energy equation gives 3*cos(theta) = 5/2, so theta = cos⁻¹(5/6).

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