(Solved): Two wheels are securely mounted on collinear shafts, initially without touching. The first wheel t ...

Two wheels are securely mounted on collinear shafts, initially without touching. The first wheel turns with angular velocity \( \omega \) while the second wheel is stationary. Both wheels are uniform disks of thickness \( d \) and density \( \rho \). The radii of the wheels are \( 2 a \) and \( a \), respectively. (a) (2 pt) Compute the momentum of inertia of each wheel in terms of \( a, \rho \), and \( d \). What is the ratio of the two moments of inertia? Now imagine that the shafts are slowly moved until the two wheels come into contact. The axes of rotation remain collinear throughout. After a while, an equilibrium is achieved and the wheels tun without their surfaces slipping relative to each other. (b) (2 pt) Compute the final angular velocities of the wheels at the end of the process in terms of \( \omega \). (c) (2 pt) Compute the ratio of the final total kinetic energy of the wheels and the initial total kinetic kinetic energies of the wheels. (d) (2 pt) Is the kinetic energy of rotation conserved? Explain.