AP Physics C: Rotational Dynamics Problems (with calculus)

AP Physics C: Mechanics turns rotational dynamics from algebra-based formulas into calculus-based setups. The most common FRQ topic: computing moments of inertia by integration. Here’s the standard setups.

Moment of inertia: the definition

For a discrete system: I = Σ miri².

For a continuous body: I = ∫ r² dm, where r is perpendicular distance from the axis.

The trick is setting up dm in terms of position. Linear mass density λ for thin rods, surface density σ for plates, volume density ρ for solids.

Setup 1 — Thin rod about its end

Rod of mass M and length L, axis at one end.

Linear density: λ = M/L. So dm = λ dx = (M/L) dx.

Integral: I = ∫0L x² · (M/L) dx = (M/L) [x³/3]0L = (M/L)(L³/3) = ML²/3.

Setup 2 — Thin rod about its center

Same rod, axis at center. Limits run from −L/2 to L/2.

I = ∫−L/2L/2 x² (M/L) dx = (M/L) [x³/3]−L/2L/2 = (M/L)(L³/12) = ML²/12.

The rod through its center has 1/4 the moment of inertia of through its end — consistent with the parallel-axis theorem.

The parallel-axis theorem (shortcut for non-center axes)

If you know Icm for an object rotating about its center of mass, the moment of inertia about a parallel axis distance d away is: I = Icm + Md².

For the rod above: Iend = Icm + M(L/2)² = ML²/12 + ML²/4 = ML²/3. ✓ Consistent with the integral.

Standard moments of inertia to memorize

  • Thin rod about its center: ML²/12
  • Thin rod about its end: ML²/3
  • Solid disk about its axis: MR²/2
  • Solid disk about a diameter: MR²/4
  • Hoop (thin ring) about its axis: MR²
  • Solid sphere: 2MR²/5
  • Hollow sphere: 2MR²/3

The AP Physics C reference table includes these — but knowing them lets you set up rolling-motion problems faster.

Common FRQ mistakes

  1. Wrong setup of dm. Linear vs. area vs. volume density — getting the wrong dimensionality of mass element ruins the integral.
  2. Wrong limits of integration. About the end of a rod: 0 to L. About the center: −L/2 to L/2.
  3. Forgetting that I depends on the axis. A rod has different I about its end vs. its center. The problem will specify; pay attention.
  4. Missing the parallel-axis term when stacking shapes. For compound objects, compute Icm first, then shift.

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