Question

If J, E and I are the angular momentum, kinetic energy of rotation and moment of inertia respectively, then which of following is incorrect?

• A. \( E = \frac{1}{2} I \omega^2 \)
• B. \( J = I \omega \)
• C. \( E = \frac{J^2}{2I} \)
• D. \( E = JI \)

Hint:

Remember the analogy between linear and rotational motion. Kinetic Energy \( K = \frac{1}{2} m v^2 \) is analogous to \( E = \frac{1}{2} I \omega^2 \). Linear momentum \( p = mv \) is analogous to \( J = I \omega \). The relationship \( K = \frac{p^2}{2m} \) is analogous to \( E = \frac{J^2}{2I} \). This can help you quickly recall or verify these formulas.

Answer 1

The Correct Option is D

Step 1: Concept Assessment:
This question evaluates understanding of the interrelationships among rotational kinetic energy (E), angular momentum (J), moment of inertia (I), and angular velocity (\(\omega\)). The objective is to identify the invalid equation.
Step 2: Core Formulas:
Fundamental definitions are:
1. Rotational Kinetic Energy: \( E = \frac{1}{2} I \omega^2 \)
2. Angular Momentum: \( J = I \omega \)
These definitions enable derivation and validation of relationships.
Step 3: Derivation and Analysis:
Evaluation of each option:

(A) \( E = \frac{1}{2} I \omega^2 \): This is the standard formula for rotational kinetic energy. Valid.

(B) \( J = I \omega \): This is the standard formula for angular momentum of a rigid body rotating about a fixed axis. Valid.

(C) \( E = \frac{J^2}{2I} \): Derived from \( J = I \omega \implies \omega = \frac{J}{I} \). Substituting into E:
\[ E = \frac{1}{2} I \left(\frac{J}{I}\right)^2 = \frac{1}{2} I \frac{J^2}{I^2} = \frac{J^2}{2I} \] Valid.

(D) \( E = JI \): This equation is dimensionally inconsistent and not derivable from standard definitions. Invalid.
Step 4: Conclusion:
The invalid relationship is E = JI.