Find the frictional force acting on a solid cylinder

[Kaushik]

Homework Statement

A solid cylinder of mass m is wrapped with an inextensible light string and, is placed on a rough inclined plane as shown in the figure below. The frictional force acting between the cylinder and the inclined plane is?

Relevant Equations

$$T + f_{s} = mg*sin(60)$$
$$r \times T = r \times f_{r}$$

This was the answer key provided:

My questions are the following:

1. if the force required for rotational equilibrium is more than the limiting static friction, then the body will rotate aka slip over the surface. When it slips, the frictional force will be kinetic and not static, right?
2. If I use $f = 0.2mg$ (as given in the answer key), then $T = f = 0.2mg$ for rotational equilibrium. But we must also maintain translational equilibrium. But, $T + f = 0.4mg ≠ mgsin(60)$. Isn't this contradicting?

 

[BvU]

1. I agree. At t=0 you have $\mu_s mg\cos\theta$ but immediately after it is $\mu_k mg\cos\theta$. I suppose the exercise asks for t = 0
2. There is no equilibrium, so $T\ne f$.