The kinetic energy of the two rings are equal and the ratio of their angular velocity is 4:3. Find the ratio of their moment of inertia.

CONCEPT:

Moment of inertia

• It plays the same role in rotational motion as mass plays in linear motion. It is the property of a body due to which it opposes any change in its state of rest or of uniform rotation.
• The moment of inertia of a particle is given as,

\(\Rightarrow I=mr^{2}\)
where r = perpendicular distance of the particle from the rotational axis
Rotational kinetic energy

• The energy, which a body has by virtue of its rotational motion, is called rotational kinetic energy.
• A body rotating about a fixed axis possesses kinetic energy because its constituent particles are in motion, even though the body as a whole remains in place.
• Mathematically rotational kinetic energy can be written as,

\(\Rightarrow KE=\frac{1}{2}Iω^{2}\)

Where I = moment of inertia and ω = angular velocity

CALCULATION:

Given \(\frac{\omega_{1}}{\omega_{2}}=\frac{4}{3}\) and KE₁ = KE₂ = KE

• Rotational kinetic energy is given as,

\(\Rightarrow KE=\frac{1}{2}Iω^{2}\)

\(\Rightarrow KE_{1}=\frac{1}{2}I_{1}ω_{1}^{2}\)     -----(1)

\(\Rightarrow KE_{2}=\frac{1}{2}I_{2}ω_{2}^{2}\)     -----(2)

Since KE₁ = KE₂,

\(\Rightarrow \frac{1}{2}I_{1}ω_{1}^{2}=\frac{1}{2}I_{2}ω_{2}^{2}\)

\(\Rightarrow \frac{I_{1}}{I_{2}}=\frac{\omega_{2}^{2}}{\omega_{1}^{2}}\)

\(\Rightarrow \frac{I_{1}}{I_{2}}=\frac{3^{2}}{4^{2}}\)

\(\Rightarrow \frac{I_{1}}{I_{2}}=\frac{9}{16}\)

• Hence, option 2 is correct.