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

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Discussion Overview

The discussion revolves around a physics problem involving a disk supported by a massless string, focusing on determining the smallest possible tension in the string at its lowest point. The context includes concepts of tension, friction, and the mechanics of the system as described in a textbook.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a scenario involving a disk of mass M and radius R, questioning the relationship between tension at the lowest point of the string and the gravitational force acting on the disk.
  • Another participant seeks clarification on the meaning of ##\pi/2## in the context of the problem and questions whether the disk is resting in a loop of the string.
  • A participant asserts that if the disk were held up by two vertical lengths of string, the tension in each would be mg/2, but questions the assumption that tensions in the vertical lengths must be equal.
  • Another participant challenges the idea that the vertical lengths have different tensions, asking for clarification on the conditions that would lead to such a conclusion, including the angles of the strings and whether the disk is spinning.

Areas of Agreement / Disagreement

Participants express differing views on the tension in the vertical lengths of the string, with some asserting they must be equal while others suggest they may differ based on the dynamics of the system. The discussion remains unresolved regarding the conditions affecting tension in the strings.

Contextual Notes

Participants note the absence of a diagram, which may limit the understanding of the system's configuration and the forces at play.

Jzhang27143
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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.)
 
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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?
 
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?
 
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.
 

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