Why angular velocity is increasing by time and not decreasing due to friction/conservation of energy law? This first picture has the question


A ring of mass $M_1$, radius $R_1$, and rotational inertia $MR^2$ is initially sliding on a frictionless surface at a constant velocity $v_0$ to the right, as shown in the image. At time $t=0$, it encounters a surface with coefficient of friction $\mu$ and begins sliding and rotating. After travelling a distance $L$, the ring begins rolling without sliding. Express all answers to the following in terms of $M$, $R$, $v_0$, $\mu$, and fundamental constants, as appropriate.
A) Starting from Newton's second law in either translational or rotational form, as appropriate, derive a differential equation that can be used to solve for the magnitude of the following as the ring is sliding and rotating.
i) The linear velocity $v$ of the ring as a function of time $t$.
ii) The angular velocity $\omega$ of the ring as a function of time $t$
B) Derive an expression for the magnitude of the following as the ring is sliding and rotating:
i) The linear velocity $v$ of the ring as a function of time $t$.
ii) The angular velocity $\omega$ of the ring as a function of time $t$

The second picture has the answer

For setting up the integral of the function
$$\int^\omega_0 d\omega = \int^t_0(\mu g/R)dt$$
For the correct answer, $$\omega=\mu gt/R$$

My question is about the answer of b ii. The expression of angular velocity indicates that the with time the angular velocity will increase. Why is that?
Shouldn't angular velocity decrease by time passing? Due to friction and law of conservative energy?
 A: There is no angular velocity when sliding at the friction-less surface. There is angular velocity when it reaches the rougher surface that causes friction. Surely, the angular velocity must have changed from zero to non-zero - clearly is is not constant over time.
What actually happens is that friction grabs the rim at the point of contact and pulls it backwards. This causes a torque that causes angular acceleration; thus increasing angular velocity over time. 
Energy conservation still holds true. The energy for this rotational motion must be taken from the linear motion. In other words, you would expect translational kinetic energy to decrease while the rotational kinetic energy increases.
While the linear velocity is being reduced and the angular velocity is increases, soon the rotation will fit the motion of the surface. Meaning, soon the rim speed will fit that of the surface, so that they do not slide over one another anymore. Which this happens you suddenly no longer have kinetic friction causing a torque. And this is a balanced rolling motion that will in theory continue forever without speeding further up or slowing down again.
