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A disc is rotating at an angular velocity $ω_{0}$ . A constant retardation torque is applied on it to stop the disc. After a certain time at which some number of rotation of the disc have been performed so that total angle rotated is $θ_{1}$ and that only $\frac{2}{3}$ th of these rotatios will further stop the disc. Find the retarding force.

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$\frac{11 ω_{0}^{2}}{14 θ_{1}}$

$\frac{9 ω_{0}^{2}}{14 θ_{1}}$

$\frac{5 ω_{0}^{2}}{14 θ_{1}}$

$\frac{3 ω_{0}^{2}}{14 θ_{1}}$

answer is C.

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Detailed Solution

Let $n_{1}$ rotations performed up to a certain time

$n_{1} = (θ_{1} / 2 π) n_{2} = \frac{2}{3} n_{1} = \frac{2}{3} - \frac{θ_{1}}{2 π} = \frac{2}{5} θ_{1} - \frac{1}{2 π} ω_{1}^{2} = ω_{0}^{2} - 2 α θ_{1} \dots . . (1) and 0 = ω_{1}^{2} - 2 α = \frac{2}{5} θ_{1} - \dots (2)$

From eqns. (1) and (2)

$\frac{4 α θ_{1}}{5} = ω_{0}^{2} - 2 α θ_{1}; α = \frac{5 ω_{0}^{2}}{14 θ_{1}}$

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