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The cross-sections of two hollow bars made of the same material are concentric circles as shown in the figure. It is given that \(r_3>r_1\) and \(r_4>r_2\), and that the areas of the cross-sections are the same. Let \(J_1\) and \(J_2\) be the torsional rigidities of the bars on the left and right, respectively. The ratio \(J_2/J_1\) is

1. \(>1\)
2. \(<0.5\)
3. \(=1\)
4. between \(0.5\) and \(1\)

Correct answer: \(>1\)

Solution

For a hollow circular section, torsional rigidity is proportional to the polar moment of inertia \(J\), which increases when more material lies farther from the center. Since the right bar has larger outer and inner radii with the same area, its material is distributed farther out, so \(J_2>J_1\).

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