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:
Rotational kinetic energy and angular momentum are directly analogous to translational kinetic energy and linear momentum. We must identify the correct mathematical relationship between them.
Step 2: Key Formula or Approach:
Rotational kinetic energy: \(E = \frac{1}{2}I\omega^2\).
Angular momentum: \(L = I\omega\).
Substitute \(\omega = \frac{L}{I}\) into the energy equation to relate \(E, L,\) and \(I\).
Step 3: Detailed Explanation:
From the angular momentum formula, isolate angular velocity (\(\omega\)):
\[ \omega = \frac{L}{I} \] Substitute this expression into the rotational kinetic energy formula:
\[ E = \frac{1}{2}I\left(\frac{L}{I}\right)^2 \] \[ E = \frac{1}{2}I \cdot \frac{L^2}{I^2} \] \[ E = \frac{L^2}{2I} \] Now, rearrange this to solve for \(L\):
\[ L^2 = 2EI \] Take the square root of both sides:
\[ L = \sqrt{2EI} \] Step 4: Final Answer:
The correct relationship is \(L = \sqrt{2EI}\).