Moment of Inertia by Torsional Pendulum Easy Explanation

Moment of inertia, written I, is the rotational twin of mass: it measures how strongly a body resists being set spinning or stopped. Just as a heavier object is harder to push, a body whose mass sits farther from its axis is harder to turn. Its SI unit is the kilogram metre squared (kg·m²).

For irregular bodies like flywheels and rotors, no simple formula gives I, so it is measured with a torsional pendulum: the body hangs from a thin wire, is twisted slightly, and its oscillation period reveals I after the wire's stiffness is calibrated with a body of known I.

Key Concepts

• Torsional rigidity (C): restoring couple per unit twist angle, in N·m per radian; a twist θ gives couple Cθ.
• Torsional pendulum: a body suspended by a vertical wire that oscillates rotationally when twisted and released.
• Time period (T): the time for one full twist-and-return oscillation, measured with a stopwatch over many oscillations.
• Reference body: a regular body such as a solid disc with exactly known I, used to find C.
• Axis dependence: I always belongs to a stated axis; the same body has different I about different axes.

Formulas

Basic relation: T = 2π √(I / C). Squaring: T² = 4π² I / C, so I = C T² / (4π²) if C is known. Calibration: with the unknown body alone (period T₁) and with a reference disc of known I₁ added (period T₂), T₁² = 4π² I / C and T₂² = 4π² (I + I₁) / C; dividing eliminates C and gives I = I₁ T₁² / (T₂² − T₁²). For a solid disc of mass M and radius R, I₁ = ½ M R². Assumptions: small twist angle (wire stays elastic), unchanged wire rigidity between runs, negligible air damping.

Worked Example

An irregular metal casting is hung from a wire and oscillates with period T₁ = 1.50 s. A solid disc of mass 0.50 kg and radius 0.10 m is clamped to the casting, and the new period is T₂ = 1.80 s.

1. Disc's I: I₁ = ½ M R² = 0.5 × 0.50 × 0.010 = 0.0025 kg·m².
2. Square the periods: T₁² = 2.25 s² and T₂² = 3.24 s².
3. Apply I = I₁ T₁² / (T₂² − T₁²) = 0.0025 × 2.25 / (3.24 − 2.25) = 0.005625 / 0.99.
4. Divide: I = 0.00568 kg·m², about 5.7 × 10⁻³ kg·m².

Common Misconceptions

• I is not just about mass: two equal masses differ greatly if one holds mass near the rim and the other near the axis.
• C must be measured, not guessed: it comes from the reference-body run; a handbook value would make the answer unreliable.
• Big twists ruin the result: the formula needs small angles where the wire stays elastic; large twists change C.

Real-World Applications

• Designing flywheels that store rotational energy smoothly in engines and presses.
• Balancing crankshafts, turbines and washing-machine drums to avoid vibration.
• Ship and aircraft designers use I to predict rolling and pitching motion.
• Robotics engineers need each link's I to size motors and control movements accurately.

The torsional pendulum turns a hard geometry problem into timing: twist, time the swings, repeat with a known disc, and the ratio of squared periods gives the unknown I, for a body of any shape.