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: Understanding the Concept:
This question tests the fundamental relationships between rotational kinetic energy (E), angular momentum (J), moment of inertia (I), and angular velocity (\(\omega\)). We need to identify which of the given equations is not a valid relationship.
Step 2: Key Formula or Approach:
The basic definitions for rotational kinetic energy and angular momentum are:
1. Rotational Kinetic Energy: \( E = \frac{1}{2} I \omega^2 \)
2. Angular Momentum: \( J = I \omega \)
We can use these two fundamental equations to derive other relationships and check the validity of the given options.
Step 3: Detailed Explanation:
Let's examine each option:

(A) \( E = \frac{1}{2} I \omega^2 \): This is the standard definition of rotational kinetic energy. This statement is correct.

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

(C) \( E = \frac{J^2}{2I} \): We can derive this relationship. From \( J = I \omega \), we can write \( \omega = \frac{J}{I} \). Substituting this into the energy equation:
\[ E = \frac{1}{2} I \omega^2 = \frac{1}{2} I \left(\frac{J}{I}\right)^2 = \frac{1}{2} I \frac{J^2}{I^2} = \frac{J^2}{2I} \] This statement is correct.

(D) \( E = JI \): This equation is dimensionally inconsistent and does not follow from the standard definitions. This statement is incorrect.
Step 4: Final Answer:
The incorrect relationship among the given options is E = JI.