The Mad Hatter - Mass on Spring-Loaded Hemisphere · USAPhO 2021 #B3
Physics · mechanics
Problem
A frictionless hemisphere of radius R is fixed on top of a flat cylinder. One end of a spring with zero relaxed length and spring constant k (i.e. the force from the spring when stretched to length ℓ is −kℓ) is fixed to the top of the hemisphere. Its other end is attached to a point mass of mass m.
[Figure: Hemisphere with radius R, showing angle θ and dashed semicircular arc]
a. The number and nature of the equilibrium points on the hemisphere depends on the value of the spring constant k. Consider the semicircular arc shown above as a dashed line, which is parameterized by angles in the range −π/2 ≤ θ ≤ π/2. Make a table indicating the number of equilibrium points on the arc, and the number that are stable, for each range of k values. A blank table for your reference is given below. (You may need more or fewer rows than shown.)
Range of k (k_min < k < k_max)
# of Equilibria
# of Stable Equilibria
0 < k < ?
? < k < ∞
For the rest of the problem, suppose the value of k is such that the mass begins at stable equilibrium on the surface of the hemisphere at angle θ₀. The mass can move on the two-dimensional surface of the hemisphere, but a radially-inward external force prevents it from jumping off the surface.
b. At t = 0, the mass is given a speed v along a line of constant latitude θ = θ₀.
[Figure: Velocity vector v along constant latitude θ = θ₀]
i. Indicate which of the following trajectories the mass takes for a short time after t = 0 and briefly explain your reasoning. The differences between the paths are exaggerated.
[Figure: Three trajectory options labeled A, B, C around θ = θ₀]
ii. What is the total radial force (i.e., normal to the surface of the hemisphere) on the mass at t = 0? Express your answer in terms of m, v, R, g, and θ₀.
c. A cylinder of radius r << Rθ₀ is placed on top of the sphere. Suppose the mass is launched at an angle α away from the direction of the spring's displacement with kinetic energy K, as shown. What is the maximum angle α_max at which the mass can be launched such that it can still hit the cylinder? Express your answer in terms of K, m, g, θ₀, r, and R. You may assume K is large enough for the mass to reach the cylinder for α = 0.
[Figure: View from above showing velocity v at angle α]