Question

Two bodies have their moments of inertia \(I\) and \(2I\) respectively about their axes of rotation. If their kinetic energies of rotation are equal, their angular momenta will be in the ratio

• A. \(2:1\)
• B. \(1:2\sqrt{2}\)
• C. \(1:\sqrt{2}\)
• D. \(1:2\)

Hint:

When rotational kinetic energy is fixed, angular momentum varies as the square root of moment of inertia.

Answer 1

The Correct Option is C

Step 1: Write expression for rotational kinetic energy.
\[ K = \frac{L^2}{2I} \]

Step 2: Apply condition of equal kinetic energies.
For the two bodies, \[ \frac{L_1^2}{2I} = \frac{L_2^2}{2(2I)} \]

Step 3: Simplify the relation.
\[ \frac{L_1^2}{I} = \frac{L_2^2}{2I} \Rightarrow L_2^2 = 2L_1^2 \] \[ \Rightarrow L_2 = \sqrt{2}L_1 \]

Step 4: Ratio of angular momenta.
\[ L_1 : L_2 = 1 : \sqrt{2} \]