Understanding Maximum Torque in a Hollow Shaft

To determine the maximum torque transmitted by a hollow shaft, we need to use the fundamental principles of torsion in circular shafts. The relationship between torque, shear stress, and the geometric properties of the shaft is given by the torsion formula.

The torsion formula is:

\[ \frac{T}{J} = \frac{\tau}{r} \]where:

• \(T\) is the torque transmitted by the shaft.
• \(J\) is the polar moment of inertia of the shaft's cross-section.
• \(\tau\) is the shear stress at a radius \(r\) from the center.

For a hollow shaft with external radius \(R\) and internal radius \(r\), the maximum shear stress occurs at the outer surface, i.e., at \(r = R\). The question provides the maximum allowable shear stress as \(\tau\). Thus, we set the shear stress at the outer radius \(R\) to be \(\tau\).

The formula becomes:

\[ \frac{T_{max}}{J} = \frac{\tau_{max}}{R} = \frac{\tau}{R} \]

From this, the maximum torque \(T_{max}\) is given by:

\[ T_{max} = \frac{J}{R} \tau \]

Next, we need to find the polar moment of inertia (\(J\)) for a hollow circular shaft. The polar moment of inertia for a solid circular shaft of radius \(R\) is \(J_{solid} = \frac{\pi}{2}R^4\). For a hollow shaft with external radius \(R\) and internal radius \(r\), the polar moment of inertia is the difference between the polar moment of inertia of a solid shaft of radius \(R\) and a solid shaft of radius \(r\).

The polar moment of inertia for a hollow shaft is:

\[ J_{hollow} = \frac{\pi}{2}R^4 - \frac{\pi}{2}r^4 = \frac{\pi}{2}(R^4 - r^4) \]

Now, substitute this expression for \(J\) into the formula for \(T_{max}\):

\[ T_{max} = \frac{\frac{\pi}{2}(R^4 - r^4)}{R} \tau \]

Simplifying the expression:

\[ T_{max} = \frac{\pi(R^4 - r^4)}{2R} \tau \]

So, the maximum torque transmitted by the hollow shaft is \(\frac{\pi}{2R}(R^4 - r^4)\tau\).

Comparing this derived formula with the given options, we find that it matches option 2.