Moment of Inertia and Rotational Kinetic Energy

The moment of inertia I = Σmr² measures a body’s resistance to changes in its rotation about an axis, depending on how its mass is distributed relative to the axis, and the rotational kinetic energy of a rotating body is E_k = ½Iω². In brief: the farther the mass is from the axis, the larger the moment of inertia. Standard results: thin ring MR², solid disc ½MR², solid sphere (2/5)MR², thin rod about its center (1/12)ML², and about an end (1/3)ML². The parallel-axis theorem gives I = I_cm + Md² for an axis parallel to one through the center of mass at a distance d.

Flywheels store energy, figure skaters spin faster by pulling their arms in, and a hollow tube rolls down a ramp more slowly than a solid cylinder. All of these depend on the moment of inertia. This guide defines the moment of inertia and rotational energy, gives the formulas for common shapes and the parallel-axis theorem, compares the shapes in a table, works through numerical examples and corrects common mistakes.

What is the moment of inertia?

In rotation, the moment of inertia I plays the role that mass plays in linear motion. For a system of point masses, I = Σm_ir_i², where r_i is the perpendicular distance of mass m_i from the axis. For continuous bodies, the sum becomes an integral, I = ∫r² dm. The SI unit is kg·m². The moment of inertia depends on the axis chosen: a rod is easier to rotate about its own axis than end over end. A body with its mass concentrated near the axis (a solid disc) has a smaller I than one with the same mass far from the axis (a ring or a bicycle wheel with a heavy rim). Racing wheels are made light to reduce I as well as mass, and flywheels have a heavy rim to store energy.

What are the values for common shapes?

For uniform bodies of total mass M (R = radius, L = length):

• thin hoop or ring about its central axis: MR²;
• solid cylinder or disc about its central axis: ½MR²;
• hollow cylinder with inner radius R₁ and outer R₂: ½M(R₁² + R₂²);
• solid sphere about a diameter: (2/5)MR²;
• thin spherical shell: (2/3)MR²;
• thin rod about its center, perpendicular: (1/12)ML²; about one end: (1/3)ML².

What is the parallel-axis theorem?

If I_cm is the moment of inertia about an axis through the center of mass, the moment of inertia about a parallel axis a distance d away is I = I_cm + Md². For a rod: I_end = (1/12)ML² + M(L/2)² = (1/12 + 1/4)ML² = (1/3)ML², in agreement with the value above. For a body made of several parts, the moments of inertia of the parts about the same axis are added.

What is rotational kinetic energy?

A particle of mass m at distance r moving with speed v = ωr has kinetic energy ½mv² = ½mr²ω². Summing over all the particles, the kinetic energy of rotation is E_k = ½Iω², the analog of ½mv². For a body that both translates and rotates (like a rolling wheel), the total kinetic energy is ½Mv_cm² + ½I_cmω². Flywheels store energy this way: a flywheel of I = 5.0 kg·m² at 3,000 rpm (314 rad/s) stores ½ × 5.0 × 314² = 247 kJ.

Moments of inertia compared

Worked example: calculations

1. Point masses. Two masses of 2.0 kg each are fixed at the ends of a light rod, 0.50 m from the axis (which passes through the center of the rod): I = 2 × 2.0 × 0.50² = 1.0 kg·m². If the masses are moved to 1.0 m from the axis, I = 4.0 kg·m² (four times larger).

2. Disc. A solid disc of mass 4.0 kg and radius 0.30 m: I = ½MR² = 0.5 × 4.0 × 0.090 = 0.18 kg·m². Rotating at 10 rad/s, E_k = ½ × 0.18 × 100 = 9.0 J. A ring of the same mass and radius has I = 0.36 kg·m² and twice the energy at the same ω.
3. Rod. A uniform rod of mass 3.0 kg and length 1.2 m. About the center: I = (1/12) × 3.0 × 1.44 = 0.36 kg·m². About one end: I = (1/3) × 3.0 × 1.44 = 1.44 kg·m². Check with the parallel-axis theorem: 0.36 + 3.0 × 0.60² = 0.36 + 1.08 = 1.44 ✓.

4. Energy in a flywheel. I = 5.0 kg·m², 3,000 rpm = 314 rad/s: E_k = ½ × 5.0 × 314² = 2.47 × 10⁵ J = 247 kJ. This can power a 1.0 kW load for 247 s (about 4 minutes) if all the energy is extracted.
5. Hammer and a barbell. A person’s arm with a mass held at the end: increasing the distance of a mass from the shoulder from 0.5 m to 1.0 m increases I fourfold, so the effort to swing it at a given angular acceleration is four times larger.

How was it introduced?

Leonhard Euler introduced the moment of inertia, and the term “moment of inertia” (momentum inertiae) appeared in his Theoria motus corporum solidorum (1765). Christiaan Huygens had studied the center of oscillation of pendulums, which is related to I. The parallel-axis theorem is named after Jakob Steiner. Today the moment of inertia is a basic quantity in engineering, used for designing flywheels, wheels, turbines and spacecraft.

Which mistakes are common?

Mistake 1: Treating the moment of inertia as a property of the body alone

It also depends on the axis. State the axis.

Mistake 2: Using the distance from the center instead of the axis

r is measured perpendicular to the axis of rotation.

Mistake 3: Forgetting the parallel-axis term

For an axis not through the center of mass, add Md².

Mistake 4: Using standard formulas for non-uniform bodies

The standard formulas assume uniform density.