Spring 2025 Challenge

A tiny pebble of mass 2M is placed on top of a frictionless hemisphere of mass M, which rests on a table (see Figure). The pebble is given a tiny kick and slides down the hemisphere. What is the minimum value of the coefficient of static friction between the hemisphere and the table that ensures the hemisphere stays at rest?  

A circular wire frame of radius a and mass M has total resistance R. It slides along a horizontal frictionless surface with constant velocity toward a region with a uniform magnetic field B pointing downward perpendicular to the plane (see Figure). As the frame transitioned into the magnetic field region, its velocity decreased by ΔV. Find the change in velocity ΔV of the frame – i.e., express it in terms of given quantities (a, M, R, B).