Rotational Kinetics — Energy of a Rotating Body A rigid body with mass moment of inertia I about a fixed axis rotates at angular velocity ω. What is its rotational kinetic energy?

Introduction / Context:
The kinetic energy of rotation is central to dynamics, energy methods, and power computations in machines (flywheels, rotors, gears). It parallels the translational form (1/2 m v^2) with angular analogs I and ω.

Given Data / Assumptions:

• Rigid body rotating about a fixed axis with angular speed ω.
• Mass moment of inertia about that axis is I.

Concept / Approach:
For an elemental mass dm at radius r with tangential speed v = ω r, the elemental kinetic energy is dK = (1/2) dm * v^2 = (1/2) dm * (ω^2 r^2). Integrating over the body yields the total rotational kinetic energy.

Step-by-Step Solution:

Verification / Alternative check:
For a point mass m at radius r, I = m r^2 and K = (1/2) m (ω r)^2 = (1/2) I ω^2, confirming the general form.

Why Other Options Are Wrong:
I ω has units of angular momentum; I ω^2 lacks the 1/2 factor; 0.5 I ω is dimensionally inconsistent (energy would require ω^2).

Common Pitfalls:
Confusing torque power P = T ω with kinetic energy; omitting the square on ω; mixing mass and area moments of inertia.

Final Answer:
0.5 I ω^2.