Hoop with backward spin returning due to friction

[Wen]

I have been stuck at this qns for 2 hours.

A gymnast often toss a hoop forward while giving it a backward spin. the hoop has a radius R and mass M, and is thrown forward with speed V. Find the minimum backward angular velocity w the gymnast must give in order to make sure it spin back eventually . Take both its static and kinectic coefficient of friction to be #.

Since friction is acting , its nett torque is not 0. Hence conserv. of angular momentum cannot be applied.

if its nett torque is Mg#R, how should i relate this to the translational and rotational K.E.?

Please teach me how to solve this.

 

[mukundpa]

if the torque is taken about contact point(bottum) then torque due to all three forces is zero and we can conserve angular momentum.

 

[Wen]

so 1/2 I w^2= 1/2 MV^2 ?

 

[Nenad]

yes, the kinetic energy of a rolling object is, as you have stated:
$$KE_{rolling} = \frac{1}{2}mv^2 + \frac{1}{2}I{\omega}^2$$

if the object is on the veeeeeeerge of rolling backwards, or back to the thrower, the inequality you have stated works.

Regards,

Nenad