Short on time? This is the recap for the calculus-based course, where you derive rotational inertias by integration. For figures and full derivations, use the full study guide.
Recap
- Rotational inertia I measures a rigid system’s resistance to changes in its rotation about a given axis.
- It depends on the mass and on how far that mass is from the axis.
- Point masses: add mr² for each, all about the same axis.
- Continuous objects: I = ∫r² dm. Pick dm so that all of it is the same distance from the axis, write dm = λ dx or σ(2πr dr), integrate, then replace λ or σ by M.
- Parallel axis theorem: I = I_cm + Md², so I is smallest about the centre-of-mass axis.
Key relationships
| Object and axis | I | How you get it |
|---|---|---|
| Point mass at distance r | mr² | definition |
| Uniform rod, perpendicular axis at x = a | M(L²/3 − La + a²) | ∫(x − a)²(M/L) dx from 0 to L |
| Uniform rod, centre | ML²/12 | a = L/2 |
| Uniform rod, end | ML²/3 | a = 0 |
| Thin hoop or cylindrical shell, central axis | MR² | all mass at R |
| Uniform disk or solid cylinder, central axis | ½MR² | rings, dm = σ 2πr dr |
| Annular ring, central axis | ½M(R₁² + R₂²) | rings from R₁ to R₂ |
| Any object, parallel axis a distance d from the cm axis | I_cm + Md² | parallel axis theorem |
Unit: kg·m².
What you must derive
Rods (uniform or nonuniform) about any perpendicular axis; thin cylindrical shells, disks and annular rings, or other coaxial rings and shells, about the central axis. Other shapes, such as spheres, are given if needed.
Mistakes to avoid
- Distance to the origin instead of to the axis. For the y-axis, r = |x|.
- r instead of r².
- A bad element. A strip across a disk is not all at one distance from the centre.
- Leaving λ₀ or σ in the answer when the question asks for M.
- Parallel axis from a non-cm axis. Always go through I_cm.
- “Hoop and disk of the same M and R have the same I.” The hoop’s is twice as big.
Quick self-check
- Two 0.30 kg beads sit 0.20 m from an axis, one on each side. I? (2 × 0.30 × 0.20² = 0.024 kg·m²)
- A uniform 0.60 kg rod, 0.80 m long, turns about one end. I? (0.60 × 0.80²/3 = 0.128 kg·m²)
- A uniform 2.0 kg disk of radius 0.15 m turns about a perpendicular axis through its rim. I? (½MR² + MR² = 1.5 × 2.0 × 0.15² = 0.0675 kg·m²)
Next: practice questions.