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

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): \(ω ∝ n^{\frac 13}\)