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The maximum angular velocity with which a thin rod can be rotated about an axis passing through its centre c and perpendicular to the length without causing rupture is $' ω'$ . Then what is the safe angular velocity with which the same rod can be rotated about an axis passing through one end and perpendicular to its length?

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$\frac{ω}{2}$

$\frac{ω}{4}$

$\frac{ω}{8}$

$\frac{ω}{\sqrt{2}}$

answer is C.

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Detailed Solution

In the first case maximum stress is induced at the mid point C and this stress is given by

$σ_{\max} = \frac{(\frac{m}{2}) ω^{2} . (\frac{l}{4})}{A} = \frac{(\frac{Alρ}{2}) ω^{2} \frac{l}{4}}{A} = \frac{1}{8} {ρω}^{2} l^{2}$

In the second case, let the angular velocity be $ω_{1}$ .

$\begin{array}{l} σ_{\max} = \frac{(ρAl) ω_{1}^{2} (\frac{l}{2})}{A} = \frac{1}{2} {ρω}_{1}^{2} l^{2} \\ ∴ \frac{1}{2} {ρω}_{1}^{2} l^{2} = \frac{1}{8} {ρω}^{2} l^{2} \\ \Rightarrow ω_{1} = \frac{ω}{2} \end{array}$

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