Fred & Barney's Car - Kinetic Energy Analysis · RMPH 2013 #1
Physics · mechanics
Problem
Fred and Barney built a car having as "wheels" two identical square prisms. The wheels are perfectly suited to the road's shape (that is a periodical repetition of identical bumps), so that the center of mass of the car does not move vertically during the trip. The wheels never slip on the road. During the car's motion, the wheel's long edge always touches a "valley" of the road; road "heights" are reached by a line passing through the half of a rectangular face of the "wheels". The vehicle's movement starts in the tops of two road bumps. Initially, the horizontal translational velocity is v0.
The total weight of the vehicle excluding its "wheels" is M⋅g, distributed equally on the two wheel axles. Further, assume that the only forces acting on car are gravity and normal force.
Task No. 1 - Kinetic energy of the car
1.a. Determine the expression of J - the moment of inertia of wheel reported at its own axles if its mass is m and the length of its square side is 2a.
1.b. Determine the expression of kinetic energy of the car and the expression of the angular velocity of its wheels:
i. when the car pass on the tops of the bumps;
ii. when the car pass with each wheel through a "valley".