Particle Motion in a Parabolic Bowl · USAPhO 2011 #B2
Physics · mechanics
Problem
A particle is constrained to move on the inner surface of a frictionless parabolic bowl whose cross-section has equation z=kr2. The particle begins at a height z0 above the bottom of the bowl with a horizontal velocity v0 along the surface of the bowl. The acceleration due to gravity is g.
[Figure: Cross-section showing parabolic bowl with equation z = kr², with vertical z-axis and horizontal r-axis]
a. For a particular value of horizontal velocity v0, which we will name vh, the particle moves in a horizontal circle. What is vh in terms of g, z0, and/or k?
b. Suppose that the initial horizontal velocity is now v0>vh. What is the maximum height reached by the particle, in terms of v0, z0, g and/or k?
c. Suppose that the particle now begins at a height z0 above the bottom of the bowl with an initial velocity v0=0.
i. Assuming that z0 is small enough so that the motion can be approximated as simple harmonic, find the period of the motion in terms any or all of the mass of the particle m, g, z0, and/or k.
ii. Assuming that z0 is not small, will the actual period of motion be greater than, less than, or equal to your simple harmonic approximation above? (You need not calculate the new value explicitly, but you should show some work to defend your answer.)