Find Max Torque Using Smaller Rad of Shaft

[1question]

Homework Statement

The question wants me to find the maximum torque.
I do not understand why the solution uses the radius of the smaller part of the shaft rather than the radius of the thicker part of the shaft. Please see the attached image.

Homework Equations

shear stress = (max torque)(maximum radius)/J where J = polar moment of inertia.

J = π/2c^4 where c = max radius of the rod

The Attempt at a Solution

I want to use 37.5mm as the radius (radius of larger part of the shaft), but it is not correct. The formula above directly relates torque T and c (radius), which is why I would assume it would be correct to use the largest c value possible. I would appreciate any insight. Thanks.

 

[Simon Bridge]

I do not understand why the solution uses the radius of the smaller part of the shaft rather than the radius of the thicker part of the shaft.

Probably because the diagram shows the torque being applied to the thin part of the shaft.

It's shear stress you are calculating - the bigger part of the shaft may get a torsional (corkscrew) and a centrifugal stress - but to get a shear stress, something has to be pulling on the surface.
I suspect you are just taking your equation too literally - go back to the definition of a shear force and see how that applies to a torque. i.e. what if you applied the torque to the inner surface of a hollow shaft?

 

[1question]

Probably because the diagram shows the torque being applied to the thin part of the shaft.

Should have seen that, haha. Thank you.

 

[SteamKing]

The smaller radius shaft must transmit the same torque as the larger radius part.

 

[rezeew]

The formula for calculating shear stress is indeed correct, however, it is important to note that the maximum torque is dependent on the maximum radius of the shaft. In this case, the smaller radius is being used because it is the point at which the torque is greatest. This is due to the fact that the smaller radius experiences a greater force due to the torque, resulting in a higher shear stress and therefore a higher maximum torque. It is important to consider the entire system and not just one component when calculating maximum torque. Additionally, using the smaller radius allows for a more conservative estimate of the maximum torque, ensuring the safety and reliability of the system.