Question

Two bodies rotate with kinetic energies 'E$_1$' and 'E$_2$'. Moments of inertia about their axis of rotation are 'I$_1$' and 'I$_2$'. If $I_1 = \frac{I_2}{3}$ and $E_1 = 27E_2$, then the ratio of angular momenta 'L$_1$' to 'L$_2$' is

• A. 1 : 3
• B. 3 : 1
• C. 1 : 1
• D. 2 : 1

Hint:

Using the formula $L = \sqrt{2IE}$ makes this a breeze. Simply multiply the two given scale factors together inside a square root: $\sqrt{\frac{1}{3} \times 27} = \sqrt{9} = 3$. This keeps the whole problem down to a single rapid step!

Answer 1

The Correct Option is B

The rotational kinetic energy ($E$) of a rotating body can be written in terms of its angular momentum ($L$) and moment of inertia ($I$) as: $$E = \frac{L^2}{2I} \implies L = \sqrt{2IE}$$ Let's set up the ratio of the angular momenta for the two bodies: $$\frac{L_1}{L_2} = \sqrt{\frac{I_1 \cdot E_1}{I_2 \cdot E_2}}$$ Given parameters: * $I_1 = \frac{I_2}{3} \implies \frac{I_1}{I_2} = \frac{1}{3}$ * $E_1 = 27E_2 \implies \frac{E_1}{E_2} = 27$ Substituting these ratios directly into the expression: $$\frac{L_1}{L_2} = \sqrt{\left(\frac{1}{3}\right) \times 27} = \sqrt{9} = \frac{3}{1}$$
Final Answer:
The ratio of the angular momenta $L_1 : L_2$ is 3 : 1, which corresponds to option (B).