Work done by friction if a body is not purely rolling

After making the diagram , I realized there is no way to know whether friction will do work or not, unless more information is provided.

The net velocity$hat i $ is given by : $v_0 - (vcostheta+ omega r) $

Friction won't do any work if the above sum is 0.

Since you mentioned , that the body is not in pure roll, It should indicate that the the net sum is not 0 and the friction will do some work as there is a relative displacement between point of contact and the ground.

I hope this helps.

In the case of no-sliding, there is a one to one mapping between the points of the surface of the roll and the ground. After any length $l$ of the circunference of the roll, corresponds the same length $l$.

In this case of braking sliding, there is a difference $Delta l = l_g - l_r$, that is, the length on the ground is greater that on the roll. The work is the product of the friction force by that length difference (that is the path of the force).

$Delta w = F_f Delta l$

The friction is the net force in the roll, if we neglect air drag. So, we can write:

$int_a^b F_f Delta l = frac{1}{2}mv^2|_a^b + frac{1}{2}Iomega^2|_a^b$

The work done by the net force corresponds to the change of the kinetic energy. In the formula, $v$ is the velocity of the center of mass, $m$ the mass of the roll, $I$ its moment of inertia, and $omega$ the angular velocity.

Sliding friction does work on the sliding body because there is relative displacement between the sliding body and the surface upon which it slides; the point if contact does move (slide) before rotating away from the surface. (Pure rolling friction does no work because there is no relative displacement.)

For a rigid body there is no dissipation of energy within the rigid body due to friction, so the force of sliding friction changes the kinetic energy of the sliding body without heating effects. In reality no body is perfectly rigid and there are heating effects.

If you search this exchange for "sliding friction" you will find detailed discussions that will provide more details.