Fisika Study Center

This problem is taken from the second IPhO Budapest, Hungary, Theoretical problems, in 1968.

Problem 1
On an inclined plane of 30° a block, mass m₂ = 4 kg, is joined by a light cord to a solid cylinder, mass m₁ = 8 kg, radius r = 5 cm (Fig. 1). Find the acceleration if the bodies are released. The coefficient of friction between the block and the inclined plane μ = 0.2. Friction at the bearing and rolling friction are negligible.

Solution
If the cord is stressed the cylinder and the block are moving with the same acceleration a. Let F be the tension in the cord, S the frictional force between the cylinder and the inclined plane (Fig. 2). The angular acceleration of the cylinder is a/r. The net force causing the acceleration of the block:



and the net force causing the acceleration of the cylinder:



The equation of motion for the rotation of the cylinder:



(I is the moment of inertia of the cylinder, S⋅r is the torque of the frictional force.) Solving the system of equations we get:





The moment of inertia of a solid cylinder is



. Using the given numerical values:



Sources/Literatures :
-Waldemar Gorzkowski, Institute of Physics, Polish Academy of Sciences, Warsaw, Poland
-The Olympic home page www.jyu.fi/ipho