Question

A constant power is supplied to a rotating disc. The relationship between the angular velocity ($\omega$) of the disc and number of rotations $n$ made by the disc is governed by:

• A. $\omega \propto n^{\frac{1}{3}}$
• B. $\omega \propto n^{\frac{2}{3}}$
• C. $\omega \propto n^{\frac{3}{2}}$
• D. $\omega \propto n^2$

Answer 1

The correct option is A $\omega \propto n^{\frac{1}{3}}$

Rotational power $=\frac{d}{dt}$ (Rotational energy)

$=\frac{d}{dt}\left(\frac{1}{2}I{\omega }^2\right)$, where
$I=$ moment of inertia
$\omega =$ angular velocity
Now, rotational power $=$ constant (given)
So, $\frac{d}{dt}\left(\frac{1}{2}I{\omega }^2\right)=k$ (constant)

or, $\int d\left(\frac{1}{2}I{\omega }^2\right)=\int kdt$

or, $\frac{1}{2}I{\omega }^2=kt$

So, ${\omega }^2\propto t$ ........(1)

But $\omega =\frac{2\pi n}{t}$ ......... (2)

Comparing (1) & (2) we get,
${\omega }^2\propto \frac{n}{\omega }$

or, ${\omega }^3\propto n$

or, $\omega \propto n^{\frac{1}{3}}$