rolling with slipping

Question

A spherical shell of mass M and radius R rolls with slipping down an incline of length D as shown in Figure 7. Icm = 2/3MR^2 .Angle of the inclined plane is θ and the coefficient of kinetic friction is  μ. When the shell reaches the bottom, find

a) the translational kinetic energy of the shell.

b) the rotational kinetic energy of the shell.

c) the amount of energy converted into heat by friction.

There was a problem while adding figure seven, but you can search and find the frigure which I think it does not neccesarry.

Answer 1

The translational KE = m g D (sin(θ) - μ cos(θ))

The rotational KE = μ² Dmg cos²(θ) ( R² m / Icm) /(sin(θ) - μ cos(θ))

The amount of energy converted to heat is equal to the change in system energy that is

heat = translational KE + rotational KE - m g D sin(θ) ( will be negative for small μ)