Question

A solid sphere of ice and a solid sphere of plastic are placed at the top of an inclined plane of length L and simultaneously released from rest, as shown in (Figure 1). Each has mass m and radius R. Assume that the coefficient of friction between the ice sphere and the incline is zero and that the plastic sphere rolls down the incline without slipping. Derive expressions for the following quantities in terms of L, θ, m, and R: (a) the total kinetic energy of each sphere at the bottom of the incline; (b) the translational speed of each sphere at the bottom of the incline; (c) the translational kinetic energy of each sphere at the bottom of the incline; (d) the rotational kinetic energy of each sphere at the bottom of the incline.

1. Write an expression for the kinetic energy of the ice sphere in terms of its translational speed v.

Express your answer in terms of some or all of the variables m, R, and v.

2. Write an expression for the kinetic energy of the plastic sphere in terms of its translational speed v.

Express your answer in terms of some or all of the variables m, R, and v.

Answer 1