§10.4 — Rotational Kinetic Energy

A spinning football thrown through the air has two types of kinetic energy:

1. Translational kinetic energy: from the motion of its center of mass
2. Rotational kinetic energy: from spinning around itself

Both are real energy — both can do work. This section explores rotational energy.

Rotational Kinetic Energy

For an object spinning in place (but center of mass at rest):

\[ K_{\text{rot}} = \frac{1}{2}I\omega^2 \]

where:

• \( I \) = moment of inertia (kg·m²)
• \( \omega \) = angular velocity (rad/s)

Parallel with Linear Energy

\[ K_{\text{linear}} = \frac{1}{2}m v^2 \]

\[ K_{\text{rot}} = \frac{1}{2}I\omega^2 \]

Substitution:

• \( m \rightarrow I \)
• \( v \rightarrow \omega \)

Exactly the same form!

Example

A cylinder of mass 2 kg and radius 0.3 m sits on a frictionless table, spinning at angular velocity ω = 5 rad/s.

Rotational energy:

\[ I_{\text{cylinder}} = \frac{1}{2}MR^2 = \frac{1}{2} \times 2 \times (0.3)^2 = 0.09 \text{ kg·m}^2 \]

\[ K_{\text{rot}} = \frac{1}{2} \times 0.09 \times 5^2 = \frac{1}{2} \times 0.09 \times 25 = 1.125 \text{ J} \]

Work and Power in Rotation

Rotational Work

Just as linear work = force × displacement:

\[ W_{\text{linear}} = F \cdot d \]

Rotational work = torque × angle:

\[ W_{\text{rot}} = \tau \cdot \theta \]

(if torque is constant and angular displacement is \( \theta \))

Unit: joule (J) — same as linear work

Example

A constant torque τ = 12 N·m is applied to a disk for 4 complete revolutions.

Work:

Angle: θ = 4 × 2π = 8π rad

\[ W = \tau \cdot \theta = 12 \times 8\pi \approx 12 \times 25.1 = 301 \text{ J} \]

Rotational Power

Power = work ÷ time

\[ P = \frac{W}{t} = \frac{\tau \theta}{t} = \tau \omega \]

where \( \omega = \theta/t \) is the average angular velocity.

Example:

An engine provides 150 N·m torque at 50 rad/s.

\[ P = 150 \times 50 = 7500 \text{ W} = 7.5 \text{ kW} \]

Note: Motor catalogs typically list power, not torque. Power depends directly on speed!

Work-Energy Theorem for Rotation

Just like linear motion:

\[ W_{\text{net}} = \Delta K_{\text{rot}} \]

Net work = change in rotational kinetic energy

Example

A disk starts from rest. A torque of 6 N·m is applied for 8 revolutions. Moment of inertia is \( I = 0.5 \) kg·m².

Work:

\[ W = 6 \times (8 \times 2\pi) = 6 \times 16\pi \approx 301 \text{ J} \]

Final rotational energy:

\[ K_{\text{rot,final}} = K_{\text{rot,initial}} + W = 0 + 301 = 301 \text{ J} \]

Final angular velocity:

\[ K_{\text{rot}} = \frac{1}{2}I\omega^2 \implies 301 = \frac{1}{2} \times 0.5 \times \omega^2 \]

\[ \omega^2 = \frac{301}{0.25} = 1204 \implies \omega \approx 34.7 \text{ rad/s} \]

Translation + Rotation

If an object moves forward (like a rolling wheel) and spins, total kinetic energy is:

\[ K_{\text{total}} = K_{\text{trans}} + K_{\text{rot}} = \frac{1}{2}m v_{\text{cm}}^2 + \frac{1}{2}I_{\text{cm}}\omega^2 \]

where \( v_{\text{cm}} \) is the velocity of the center of mass and \( I_{\text{cm}} \) is the moment through its center.

Example

A wheel: mass 5 kg, radius 0.4 m, center-of-mass speed 3 m/s, angular velocity ω = 10 rad/s (rolling).

Translational energy:

\[ K_{\text{trans}} = \frac{1}{2} \times 5 \times 3^2 = 22.5 \text{ J} \]

Rotational energy:

\[ I = \frac{1}{2} \times 5 \times (0.4)^2 = 0.4 \text{ kg·m}^2 \]

\[ K_{\text{rot}} = \frac{1}{2} \times 0.4 \times 10^2 = 20 \text{ J} \]

Total:

\[ K_{\text{total}} = 22.5 + 20 = 42.5 \text{ J} \]

Note: About 47% of energy is in rotation! This matters for spinning balls.

Energy Conservation with Friction and Loss

Rotational energy converts to:

• Heat: via friction
• Linear kinetic energy: in collisions
• Potential energy: when climbing uphill

Example: A tennis ball thrown with heavy spin. Air friction slows it—rotational energy dissipates as sound and heat. This is behind the Magnus effect.

Comparison: Linear vs Rotational Energy

Aspect	Linear	Rotational
Energy	\( K = \frac{1}{2}mv^2 \)	\( K = \frac{1}{2}I\omega^2 \)
Work	\( W = F \cdot d \)	\( W = \tau \cdot \theta \)
Power	\( P = F \cdot v \)	\( P = \tau \cdot \omega \)
Force/Torque	\( F = ma \)	\( \tau = I\alpha \)

What You Should Know

• Rotational kinetic energy: \( K_{\text{rot}} = \frac{1}{2}I\omega^2 \)
• Rotational work: \( W = \tau \theta \)
• Rotational power: \( P = \tau \omega \)
• Work-energy theorem: \( W_{\text{net}} = \Delta K_{\text{rot}} \)
• Translation + rotation: sum both energies
• Parallel with linear quantities

Preview of §10.5

So far we've assumed an object either purely translates or purely rotates. In reality, many objects do both — they roll. When a ball or wheel rolls without slipping, there's a geometric constraint linking linear and rotational motion. What does this constraint mean for energy?

📚 See also: Halliday Vol 1, Ch 10, §10.4 — Rotational kinetic energy. 🔗 Reference: §7.6 (Power) — linear work and power for comparison.

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