Question about torsional shear stress

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In the discussion about torsional shear stress in a circular rod, the key point is that when a moment (M) is applied to one end and a smaller moment (M/3) to the other, the rod is free to rotate, leading to an unbalanced moment. This unbalanced moment causes the rod to accelerate, meaning the effective moment for calculating torsional shear stress is not simply M. The maximum moment that contributes to torsion is determined to be M/3, as this is the moment that balances the acceleration of the rod. Clarification is needed on the interpretation of moments applied at both ends to fully understand the stress distribution. The discussion emphasizes the importance of equilibrium in analyzing torsional effects.
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If a circular rod is allowed to rotate freely and a moment (M) is applied to one end of the rod. A moment acting in the opposite direction, with magnitude (M/3), is applied to the other end of the rod. What is the maximum moment in the rod that would be used to calculate the maximum torsional shear stress in the equation (T*c)/J?
 
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Welcome to PF.

Because the rod is free to rotate, the unbalanced moment will result in an acceleration of the rotation of the rod. The rod will be subjected only to the moment (M), which will be balanced by (M/3) plus the moment causing acceleration.
 
Welcome to PF.

rcw2004 said:
If a circular rod is allowed to rotate freely and a moment (M) is applied to one end of the rod. A moment acting in the opposite direction, with magnitude (M/3), is applied to the other end of the rod. What is the maximum moment in the rod that would be used to calculate the maximum torsional shear stress in the equation (T*c)/J?
Please show your work based on the hints given by Baluncore. Thank you.
 
Baluncore said:
Welcome to PF.

Because the rod is free to rotate, the unbalanced moment will result in an acceleration of the rotation of the rod. The rod will be subjected only to the moment (M), which will be balanced by (M/3) plus the moment causing acceleration.
Thank you for the answer. I am still confused, is M not the moment that is causing the acceleration? My initial thought was that one third of M would go towards resisting the M/3 moment, and the other two thirds of M would accelerate the rod. Then the Moment that is actually twisting the rod and causing torsion would be M/3. Is this incorrect? and if so why?
 
rcw2004 said:
Is this incorrect? and if so why?
The moment, M, is applied to one end of the rod.
The rod must be able to transmit that M, along the rod, to where it will be countered by the 2M/3 that accelerates the rod, and the far end M/3.

There are multiple interpretations of your question. We need a diagram, or the original wording to determine the situation.
 
Let's say you apply M/3 to both ends - then the stress is M/3 times c/J. Since one end applies M then there is no equilibrium and rod accelerates since free to rotate. So max twist moment is M/3
 
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