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1) Which of the following forces is conservative?
(a) F = k (x, 2y, 3z) where k is constant.
(b) F = k (y, x, 0).
(c) F = k (-y, x, 0).
For those which are conservative, find the corresponding potential energy U, and verify by direct differentiation that F = -ΔU.
2) The figure shows a child's toy, which has the shape of a cylinder mounted on top of a hemisphere. The radius of the hemisphere is R and the CM of the whole toy is at height h above the floor.
(a) Write down the gravitational potential energy when the toy is tipped to an angle θ from the vertical.
(b) For what values of R and h is the equilibrium θ = 0 stable?
Using the Hamiltonian for the harmonic oscillator, or otherwise, show that the expectation value of the kinetic energy is equal to the expectation value of the potential energy and half that of the total energy.
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