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Chapter 5: Problem 1

The solid shaft of radius \(r\) is subjected to a torque \(\mathbf{T}\). Determine the radius \(r^{\prime}\) of the inner core of the shaft that resists one-half of the applied torque \((T / 2) .\) Solve the problem two ways: (a) by using the torsion formula, (b) by finding the resultant of the shear-stress distribution.

Short Answer

Expert verified
The radius \(r'\) of the inner core of the shaft which resists half of the applied torque is given by \(r' = \frac{r}{\sqrt[4]{2}}\), irrespective of whether the torsion formula or the shear stress distribution method is used.

Step by step solution

01

Apply the Torsion formula

The torsion formula that gives the shear stress in a shaft subjected to a torque is given as \(\tau = \frac{Tr}{J}\) where \(\tau\) is the shear stress, \(T\) is the torque, \(r\) is the radius and \(J\) is the polar moment of inertia. For a solid shaft, the polar moment of inertia \(J\) can be given as \(\frac{\pi r^4}{2}\), thus the radius resisting half the torque (\(T/2\)) can be found as \(r' = \sqrt[4]{\frac{T \cdot J}{2\tau}} = \sqrt[4]{\frac{T \cdot \frac{\pi r^4}{2}}{2\tau}}\)
02

Find Shear Stress Distribution

The shear stress distribution across the radius of the shaft is linear. So, the shear stress at \(r^\prime\) would be half of the shear stress at the outer radius \(r\), Therefore, \(\tau(r') = \frac{1}{2}\tau(r) \) Since \(\tau(r) = \frac{T}{J} \cdot r = \frac{Tr}{\frac{\pi r^4}{2}} \) and \(\tau(r') = \frac{1}{2}\tau(r) = \frac{\frac{T}{2}}{J} \cdot r' = \frac{T r'}{2\cdot\frac{\pi r'^4}{2}} \implies r' = \frac{r}{\sqrt[4]{2}}\)
03

Compare results

By comparing the results from step 1 and step 2, it's evident that the resultant radius under both the situations is the same. Therefore, either method can be used to determine the radius of the shaft that will carry half the applied torque.

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Most popular questions from this chapter

The solid shaft of radius \(c\) is subjected to a torque \(\mathbf{T}\) at its ends. Show that the maximum shear strain in the shaft is \(\gamma_{\max }=T c / J G .\) What is the shear strain on an element located at point \(A, c / 2\) from the center of the shaft? Sketch the shear strain distortion of this element.

The tubular drive shaft for the propeller of a hovercraft is \(6 \mathrm{m}\) long. If the motor delivers \(4 \mathrm{MW}\) of power to the shaft when the propellers rotate at 25 rad \(/\) s, determine the required inner diameter of the shaft if the outer diameter is \(250 \mathrm{mm}\). What is the angle of twist of the shaft when it is operating? Take \(\tau_{\text {allow }}=90 \mathrm{MPa}\) and \(G=75 \mathrm{GPa}\)

The shaft is subjected to a distributed torque along its length of \(t=\left(10 x^{2}\right) \mathrm{N} \cdot \mathrm{m} / \mathrm{m},\) where \(x\) is in meters. If the maximum stress in the shaft is to remain constant at \(80 \mathrm{MPa}\) determine the required variation of the radius \(c\) of the shaft for \(0 \leq x \leq 3 \mathrm{m}\)

The 60-mm-diameter shaft is made of \(6061-\) T 6 aluminum. If the allowable shear stress is \(\tau_{\text {allow }}=80 \mathrm{MPa}\) and the angle of twist of disk \(A\) relative to disk \(C\) is limited so that it does not exceed 0.06 rad, determine the maximum allowable torque \(\mathbf{T}\)

A bar having a circular cross section of 3 in. diameter is subjected to a torque of 100 in. \(\cdot\) kip. If the material is elastic perfectly plastic, with \(\tau_{Y}=16 \mathrm{ksi}\) determine the radius of the elastic core.
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