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

Answer: (A)

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

Answer 2

Rotational energy $= \frac 12 Iω^2$

Rotational power $= \frac {d}{dt} (\frac 12 Iω^2)$

Given that Rotational power is constant,
$⇒ \frac {d}{dt} (\frac 12 Iω^2) = K$ (Constant)

$⇒ ∫d (\frac 12 Iω^2) = ∫K dt$

$⇒ \frac 12 Iω^2 = Kt$
So, $ω^2 ∝ t$ …….. (1)
But we know that,
$ω = \frac {2\pi rn}{t}$ ……. (2)
From eq (1) and (2),
$ω^2 ∝ \frac nω$
$⇒ ω^3 ∝ n$
$⇒ ω ∝ n^{\frac 13}$

So, the correct option is (A): $ω ∝ n^{\frac 13}$