A weightless rigid rod with a small iron bob at the end is hinged at point A by a ball and socket joint to the wall so that it can rotate in all directions. The rod is kept in the horizontal position

Question

A weightless rigid rod with a small iron bob at the end is hinged at point $A$ by a ball and socket joint to the wall so that it can rotate in all directions. The rod is kept in the horizontal position by a vertical inextensible string of length $20 cm$ , fixed at its midpoint. The bob is displaced slightly, perpendicular to the plane of the figure. If the period of small oscillations of the system is $\frac{x π}{10} s$ , then what is the value of $x$ ? [Take $g = 10 m s^{- 2}$ ]

TestHub · Accepted Answer

The small oscillation of the bob occur relative to a stationary axis passing through $AB$ . The line $CD$ is perpendicular to the axis $AB$ . The bob behaves as a simple pendulum of length $CD$ oscillating due to component of its weight $m g \cos θ$ .

If the length $AD$ is equal to $L$ , then the length $CD = L \sin θ$ .

$T = 2 π \sqrt{\frac{L \sin θ}{g c o s θ}}$

$T = 2 π \sqrt{\frac{L \tan θ}{g}}$

$T = 2 π \sqrt{\frac{2 l}{g}} = \frac{2 π}{5} = \frac{4 π}{10}$

$\Rightarrow x = 4$

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