First slide

Stress and strain

Question

A uniform rod is rotating about one of its end with a constant angular velocity $ω$ in gravity free space. The radial strain produced in two halves is $k (> 1)$ . Then find the value of $\frac{20 k}{11}$ .

Difficult

A

1
B

2
C

3
D

4

Solution

$dL = \frac{T}{AY} dx = \frac{{mω}^{2}}{2 ALY} dx ∵ T = \frac{{mω}^{2}}{2 L} (L^{2} - x^{2}) ∴ ∆ L = \frac{{mω}^{2}}{2 ALY} (L^{2} x - \frac{x^{3}}{3}) ∆ L for entire rod = k . \frac{2 L^{3}}{3} = k = constant ∆ L for first half = {kL}^{3} (\frac{1}{2} - \frac{1}{24}) = \frac{11 {kL}^{3}}{24} ∆ L for second half = {kL}^{3} (\frac{2}{3} - \frac{11}{24}) = \frac{5}{24} {kL}^{3} ∴ k = \frac{11}{5} ∴ \frac{20 k}{11} = 4$

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