First, we need to get the relation between time required to stop and dimension of the square. We do some rough calculations.
Assume length of the square's sides are d, mass is m, and coefficient of friction is μ. Hence, the torque works on the square is

Combining these two equations to get α,

Because \alpha is angular acceleration,

And now, we can get

Because is constant, we can conclude that tstop is proportional to d.

Now consider there are 4 squares rotating around the edges like shown below.
Because they all have same condition, they will stop after t time. The condition above is just the same as we rotate a large square of side 2d around its center.
Using the equation above, so we can get the time for a square with side d,

That's my solution.
If you have any question, correction, or comment, please leave a comment below.
Thank you.

Solution - Rotating SquareSocialTwist Tell-a-Friend


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