Rotational Inertia: Why Shape Beats Mass
- Explain rotational inertia as mass weighted by the square of its distance from the axis
- Compare the standard rigid-body coefficients and predict which object is harder to spin
- Show that the same object has different rotational inertia about different axes
Inertia depends on where the mass is, not just how much
Translational inertia is just mass: 3 kg resists acceleration the same way whatever its shape. Rotational inertia is different, because a particle far from the axis has to travel much further — and therefore much faster — to achieve the same angular speed as one close in. Rotational inertia weights each bit of mass by the square of its distance from the axis, I = Σmr², so moving mass twice as far out multiplies its contribution by four.
The standard coefficients, and what they mean
For common shapes rotating about their center, I = cMR², where the coefficient c encodes how the mass is distributed. A hoop or thin ring has c = 1, because every scrap of mass sits at the full radius R. A solid cylinder or disk has c = ½, since much of its mass is closer in. A solid sphere has c = 2/5, more concentrated still. A hollow sphere has c = 2/3 — between the disk and the hoop, as you would expect for a shell. A rod about its center has c = 1/12; about its end, c = 1/3, four times larger.
Same object, different axes, different I
Rotational inertia is a property of an object and an axis, never of the object alone. A meter stick spun about its middle has I = ML²/12; spun about one end it has I = ML²/3 — four times harder to start rotating, despite being the identical stick. This is why an ice skater does not change mass when pulling her arms in but does change how fast she spins, and why a long baseball bat feels heavier held at the knob than choked up.
Two objects each have mass 2.0 kg and radius 0.30 m: a solid disk and a thin hoop. The same 1.5 N·m torque is applied to each about its center. Compare their angular accelerations.
- 1.Disk: I = ½MR² = ½(2.0)(0.30)² = ½(2.0)(0.090) = 0.090 kg·m².
- 2.Hoop: I = MR² = (2.0)(0.090) = 0.18 kg·m².
- 3.Disk: α = τ/I = 1.5 ÷ 0.090 = 17 rad/s².
- 4.Hoop: α = τ/I = 1.5 ÷ 0.18 = 8.3 rad/s².
The complete translation table
Every rotational equation is a translational one with the analogous symbols swapped: x → θ, v → ω, a → α, m → I, F → τ, p → L. So v = v₀ + at becomes ω = ω₀ + αt; ½mv² becomes ½Iω²; and Fnet = ma becomes τnet = Iα. Learning the table is faster than learning the rotational equations separately, and it means every problem-solving habit you built in Units 1–4 transfers directly.
Rotational inertia has units of kg·m², not kg. Two objects of equal mass can differ in rotational inertia by a factor of several, and an answer in kilograms is a sign that a radius was dropped somewhere.
A solid disk and a thin hoop have the same mass and radius. Which requires more torque to reach a given angular acceleration?
A uniform rod of mass M and length L has I = ML²/12 about its center. About one end its rotational inertia is:
A point mass of 3.0 kg sits 2.0 m from an axis. What is its rotational inertia about that axis?
The AP equation sheet gives you the rotational inertia coefficients, so do not memorize them — but do learn their order (sphere < disk < hoop). Ranking questions ask which object wins a race or needs more torque, and the ranking alone answers them.
Answer the 3 checkpoints as you read.
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