An ideal yo-yo is made of two uniform disks of radius R connected by a cylindric

Question

An ideal yo-yo is made of two uniform disks of radius R connected by a cylindrical axle of radius r, with total mass m. The axle and disks are made of the same material. An ideal string is wrapped around the axle. We set the yo-yo on a table and pull the string with force F at angle ? above the horizontal. We find that the yo-yo quickly reaches a state in which it is simply spinning in-place at constant angular speed ? on the tabletop.

Show that the angle for which this occurs is independent of the coefficient of friction between the table and yo-yo, the mass of the yo-yo, the force with which we pull, and the speed at which the yo-yo spins.

What is the angle in terms of any of the remaining given quantities?

Explanation / Answer

The sum of the torques would equal zero:

torque due to tension in string = frictional torque
(r)(T) = (?N)(R)

horizontal forces on the yo-yo are also equal zero:

force due to tension = frictional force
Tcos? = ?N

Substitute into the first equation for frictional force:
(r)(T) = (Tcos?)(R)
cos? = r/R

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