Oscillating Blocks with Friction · USAPhO 2015 #A3
Physics · mechanics
Problem
A large block of mass mb is located on a horizontal frictionless surface. A second block of mass mt is located on top of the first block; the coefficient of friction (both static and kinetic) between the two blocks is given by μ. All surfaces are horizontal; all motion is effectively one dimensional. A spring with spring constant k is connected to the top block only; the spring obeys Hooke's Law equally in both extension and compression. Assume that the top block never falls off of the bottom block; you may assume that the bottom block is very, very long. The top block is moved a distance A away from the equilibrium position and then released from rest.
[Figure: Diagram showing bottom block mb on frictionless surface with top block mt connected to a spring with spring constant k]
a. Depending on the value of A, the motion can be divided into two types: motion that experiences no frictional energy losses and motion that does. Find the value Ac that divides the two motion types. Write your answer in terms of any or all of μ, the acceleration of gravity g, the masses mt and mb, and the spring constant k.
b. Consider now the scenario A≫Ac. In this scenario the amplitude of the oscillation of the top block as measured against the original equilibrium position will change with time. Determine the magnitude of the change in amplitude, ΔA, after one complete oscillation, as a function of any or all of A, μ, g, and the angular frequency of oscillation of the top block ωt.
c. Assume still that A≫Ac. What is the maximum speed of the bottom block during the first complete oscillation cycle of the upper block?