A smooth small groove (centre O) of radius $R = 1 m$ is cut in a horizontal table and a particle of mass $π^{2}$ kg is kept in it at $(∠ A O C = 90^{0})$ . It is attached to point C by an elastic string of force constant $100 N / m$ and natural length $π / 2$ . The block is imparted velocity of $v_{0} = 5 m / s$ along the groove, as shown. if maximum value of $θ$ is $\frac{π}{x}$ . Find the value of $x$ .

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Solution

Conserving of energy,k
K.E. = P.E. of elastic

$\frac{1}{2} m v^{2} = \frac{1}{2} k x^{2}$
$\Rightarrow \frac{1}{2} π^{2} \times 5 \times 5 = \frac{1}{2} \times 100 \times x \times x$
$\Rightarrow x^{2} = \frac{π^{2}}{4}$
$x = \frac{π}{2}$

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