Problem on torus

[Arthur]

Please help me solve this problem.

A torus of homogeneous material (having radii R and r, r \ll R, where r is the radius of the torus tube, R is the radius from the center of the hole to the center of the tube) is freely sliding on a horizontal plane, its initial velocity is v. The torus is also spinning around its own symmetry axis, its initial angular velocity is \omega_0. The coefficient of kinetic friction is \mu. The drag coefficient of the air can be neglected. How does the torus’ center of mass move? Calculate its trajectory.

[Alisher (Alisher Yerkebayev)]

Let’s describe torus’ motion as the sum of the rotation around the center O and its translational motion. Hence, the velocity of an arbitrary point A is

\vec v_A = \vec\omega\times\overrightarrow{OA} + \vec v_O.

For an infinitesimally small piece Rd\theta of the torus, the friction force is

d\vec f=-\mu mg\cdot\frac{d\theta}{2\pi}\cdot\frac{\vec v_A}{|\vec v_A|} .

Now you should sum all these small forces and find their net torque exerting on the torus.

[Arthur]

Ok thank you!