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  4. A constant power is supplied to a disc that can rotate freely about a fixed vertical axis. Angular velocity (ω ) of the disc varies with the number of rotations ( n ) made by the disc after starting from rest as ω =knx . Hence x is: (K is a constant)

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Q. A constant power is supplied to a disc that can rotate freely about a fixed vertical axis. Angular velocity $\left(\omega \right)$ of the disc varies with the number of rotations ( $n$ ) made by the disc after starting from rest as $\omega =kn^{x}$ . Hence $x$ is: (K is a constant)

NTA AbhyasNTA Abhyas 2020System of Particles and Rotational Motion

Solution:

$\tau=I\alpha $
$P=\tau\omega $
$P=I \, \alpha \times \omega $
$P=I\omega ^{2}\frac{d \omega }{d \theta }$
$\omega ^{2}d\omega \propto d\theta \Rightarrow \frac{\omega ^{3}}{3} \propto \theta $
$\omega \times \theta ^{1 / 3}$ .