Question about torsional shear stress

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The discussion centers on calculating the maximum torsional shear stress in a circular rod subjected to opposing moments. When a moment (M) is applied to one end and a moment (M/3) is applied to the opposite end, the rod experiences an unbalanced moment leading to acceleration. The maximum moment used for calculating torsional shear stress is determined to be M/3, as the rod is free to rotate and cannot achieve equilibrium under the applied forces.

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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?
 
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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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