Solution Found!
(a) For a given allowable shearing stress, determine the
Chapter 3, Problem 3.30(choose chapter or problem)
(a) For a given allowable shearing stress, determine the ratio T/w of the maximum allowable torque T and the weight per unit length w for the hollow shaft shown. (b) Denoting by \((T / w)_{0}\) the value of this ratio for a solid shaft of the same radius \(c_{2}\), express the ratio T/w for the hollow shaft in terms of \((T / w)_{0}\) and \(c_{1} / c_{2}\).
Questions & Answers
QUESTION:
(a) For a given allowable shearing stress, determine the ratio T/w of the maximum allowable torque T and the weight per unit length w for the hollow shaft shown. (b) Denoting by \((T / w)_{0}\) the value of this ratio for a solid shaft of the same radius \(c_{2}\), express the ratio T/w for the hollow shaft in terms of \((T / w)_{0}\) and \(c_{1} / c_{2}\).
ANSWER:
Step 1 of 4
Part (a):
The torque for hollow shaft can be obtained from the formula of allowable shearing stress,
\(\begin{array}{c} \tau_{m}=\frac{T c_{2}}{J} \\ T=\frac{\tau_{m} J}{c_{2}} \end{array}\)
The polar moment of inertia for hollow shaft is,
\(J=\frac{\pi}{2}\left(c_{2}^{4}-c_{1}^{4}\right)\)
For \(J=\frac{\pi}{2}\left(c_{2}^{4}-c_{1}^{4}\right)\) in equation (1),
\(\begin{array}{c} T=\frac{\left(\tau_{m}\right) \frac{\pi}{2}\left(c_{2}^{4}-c_{1}^{4}\right)}{c_{2}} \\ T=\frac{\left(\tau_{m}\right) \pi\left(c_{2}^{4}-c_{1}^{4}\right)}{2 c_{2}} \end{array}\)