This is very similar to this question, with the main difference being that it doesn't assume that $\vec v=\mathbf0$.

Assume that you have a rolling and sliding sphere with radius and mass 1 with a uniform density of 1, on a frictionless surface with linear velocity $\vec v$ and angular velocity $\vec\omega$.

At time $t=0$, the surface has a coefficient of friction of $\mu$ and coefficient of rolling friction $C_{rr}$. No energy is lost to heat or anything like that.

1. What are the linear and angular velocities of the sphere at time $t$?
2. (Less important) Assume there is another pulling force $\vec g$. What is the new answer to 1?

This is very similar to this question, with the main difference being that it doesn't assume that $\vec v=\mathbf0$.

Assume that you have a rolling and sliding sphere with radius and mass 1 with a uniform density of 1, on a frictionless surface with linear velocity $\vec v$ and angular velocity $\vec\omega$.

At time $t=0$, the surface has a coefficient of friction of $\mu$ and coefficient of rolling friction $C_{rr}$.

1. What are the linear and angular velocities of the sphere at time $t$?
2. (Less important) Assume there is another pulling force $\vec g$. What is the new answer to 1?