Moment of Inertia: The Rotational “Mass”
When an object spins around a fixed line (axis), every bit of matter in it moves in a circle. A piece at distance \(r_i\) from the axis has speed \(v_i = r_i\omega\), where \(\omega\) is the angular speed. Its kinetic energy is \[ k_i = \tfrac12 m_i v_i^2 = \tfrac12 m_i r_i^2 \omega^2 . \] Adding over all pieces gives the total rotational kinetic energy \[ K = \tfrac12 \Bigl(\sum m_i r_i^2\Bigr)\omega^2 = \tfrac12 I\,\omega^2 , \] with \[ I = \sum m_i r_i^2 , \] the moment of inertia. Just as mass measures resistance to a push in straight-line motion, \(I\) measures resistance to a twist around the chosen axis. :contentReference[oaicite:0]{index=0}
Key Features of \(I\)
- Depends on how the mass is spread out and on the chosen axis. Slide the axis or reshape the body and \(I\) changes. :contentReference[oaicite:1]{index=1}
- Units: \(\text{kg·m}^2\); dimensions \(M L^2\). :contentReference[oaicite:2]{index=2}
- Large \(I\) smooths out sudden speed changes. That is why engines carry a heavy wheel called a flywheel. :contentReference[oaicite:3]{index=3}
Quick Examples
Thin ring (radius \(R\), mass \(M\)) spinning in its own plane: every bit lies at the same distance \(R\), so \[ I = M R^2 . \] Pair of small masses on a light rod (length \(l\), total mass \(M\)) about the mid-point: \[ I = \frac{M l^2}{4}. \] :contentReference[oaicite:4]{index=4}
Table 1 — Moments of Inertia for Popular Shapes
| Body | Axis | \(I\) |
|---|---|---|
| Thin circular ring, radius \(R\) | Perpendicular through centre | \(M R^2\) |
| Thin circular ring, radius \(R\) | Diameter | \(\tfrac12 M R^2\) |
| Thin rod, length \(L\) | Perpendicular at mid-point | \(\tfrac1{12} M L^2\) |
| Circular disc, radius \(R\) | Perpendicular through centre | \(\tfrac12 M R^2\) |
| Circular disc, radius \(R\) | Diameter | \(\tfrac14 M R^2\) |
| Hollow cylinder, radius \(R\) | Cylinder axis | \(M R^2\) |
| Solid cylinder, radius \(R\) | Cylinder axis | \(\tfrac12 M R^2\) |
| Solid sphere, radius \(R\) | Diameter | \(\tfrac25 M R^2\) |
These ready-to-use formulas save time during quick calculations. :contentReference[oaicite:5]{index=5}
Radius of Gyration (\(k\))
For every shape, \[ I = M k^2 . \] Think of \(k\) as the single distance at which you could bunch all the mass without changing \(I\). Examples:
- Thin rod about its mid-point: \(k = L/\sqrt{12}\).
- Disc about a diameter: \(k = R/2\).
This idea gives an intuitive feel for how mass distribution affects rotation. :contentReference[oaicite:6]{index=6}
High-Yield Ideas for NEET
- Moment of inertia definition: \(I = \sum m_i r_i^2\).
- Rotational kinetic energy: \(K = \tfrac12 I \omega^2\).
- Standard \(I\) formulas: ring, disc, sphere, rod, cylinder (see Table 1).
- Radius of gyration: \(I = M k^2\) and handy values like \(k = L/\sqrt{12}\) for a rod.
- Flywheel application: large \(I\) in engines smooths speed and prevents jerks.
