Question

If the moment of inertia, rotational kinetic energy and angular momentum of a body are $I$, $E$ and $L$ respectively, then,

• A. $I = \frac{E}{L}$
• B. $L = EI$
• C. $E = 2IL$
• D. $L = \sqrt{2EI}$
• E. $2E = \frac{I}{L}$

Hint:

Just like linear motion has \( K = \frac{p^2}{2m} \), rotational motion has \( E = \frac{L^2}{2I} \). The formulas are identical if you swap \( p \leftrightarrow L \) and \( m \leftrightarrow I \).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The rotational motion of a rigid body is analogous to linear motion. Rotational Kinetic Energy (\( E \)) is analogous to linear Kinetic Energy (\( K = p^2/2m \)).
Key Formula or Approach:
Rotational Kinetic Energy: \( E = \frac{1}{2} I \omega^2 \)
Angular Momentum: \( L = I \omega \)

Step 2: Detailed Explanation:
From the angular momentum formula, we have:
\[ \omega = \frac{L}{I} \]
Substitute this into the expression for rotational kinetic energy:
\[ E = \frac{1}{2} I \left( \frac{L}{I} \right)^2 \]
\[ E = \frac{1}{2} I \frac{L^2}{I^2} = \frac{L^2}{2I} \]
Rearrange the equation to solve for \( L \):
\[ L^2 = 2EI \]
\[ L = \sqrt{2EI} \]

Step 3: Final Answer:
The relation is $L = \sqrt{2EI}$.