Find the Smallest Possible Tension in a Massless String Supporting a Disk

[Jzhang27143]

A disk of mass M and radius R is held up by a massless string. (The two ends of the string are connected to a ceiling and the disk rests on the bottom of the string.) The coefficient of friction between the disk is μ. What is the smallest possible tension in the string at its lowest point?

This is from "Introduction to Classical Mechanics" by David Morin. I am confused as to how T(∏/2) = Mg/2. T(∏/2) refers to the tension in the rightmost point of the disk where the string does not touch the disk anymore.)

 

[Simon Bridge]

What does the $\pi/2$ refer to?

I take it, by "in the bottom" you mean the disk rests in the loop of the string?

If the disk were just held up by two vertical lengths of string - what would the tension in each string be?

 

[Jzhang27143]

They would be mg/2. In the problem, why do the tensions in the vertical lengths have to be equal? The tension in the string increases opposite the direction of friction so from this argument, I see that the vertical lengths have different tensions. What am I missing?

 

[Simon Bridge]

How do you see that the vertical lengths have different tensions?
Are they hanging at different angles?
Is the disk spinning?

Remember - I cannot see any diagram you may be looking at.