Rotational Motion

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 ( ). 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:
2. Angular Momentum:
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) : This is the standard definition of rotational kinetic energy. This statement is correct.

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

(C) : We can derive this relationship. From , we can write . Substituting this into the energy equation:
This statement is correct.

(D) : 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.

Step 1: Understanding the Concept:
This question tests the understanding of the analogies between physical quantities in translational (linear) motion and rotational motion. Each quantity in linear motion has a corresponding analogue in rotational motion.
Step 2: Detailed Explanation:
Let's find the rotational analogue for each quantity in List-I.

(A) Force ( ): In linear motion, force causes linear acceleration ( ). The rotational analogue is Torque ( ), which causes angular acceleration ( ). Thus, (A) matches with (I).

(B) Distance covered (s): This is a linear displacement. The rotational analogue is angular displacement, or the Angle described ( ). Thus, (B) matches with (II).

(C) Mass (m): Mass is the measure of inertia, i.e., resistance to change in linear motion. The rotational analogue is the Moment of inertia (I), which is the measure of resistance to change in rotational motion. Thus, (C) matches with (III).

(D) Linear velocity ( ): This is the rate of change of linear displacement. The rotational analogue is Angular velocity ( ), which is the rate of change of angular displacement. Thus, (D) matches with (IV).
Step 3: Final Answer:
The correct matching is (A)-(I), (B)-(II), (C)-(III), and (D)-(IV). This corresponds exactly to option (A).