Moment of inertia of a thin rod about its centre
I = ML²/12, radius of gyration k = L/(2√3)
Axis: through the centre, perpendicular to its length.
Derivation
- Take a piece dx at distance x from the centre. Its mass is dm = (M/L) dx.
- I = ∫ x² dm = (M/L) ∫ x² dx from −L/2 to L/2.
- = (M/L) × [x³/3] = (M/L) × (L³/24 + L³/24) = ML²/12.
Moment of inertia of standard bodies
| Body and axis | I |
|---|
| Thin ring, through the centre, perpendicular to its plane | MR² |
| Thin ring, about a diameter | MR²/2 |
| Thin ring, about a tangent in its plane | 3MR²/2 |
| Thin ring, about a tangent perpendicular to its plane | 2MR² |
| Uniform disc, through the centre, perpendicular to its plane | MR²/2 |
| Uniform disc, about a diameter | MR²/4 |
| Uniform disc, about a tangent in its plane | 5MR²/4 |
| Uniform disc, about a tangent perpendicular to its plane | 3MR²/2 |
| Thin rod, through the centre, perpendicular to its length | ML²/12 |
| Thin rod, through one end, perpendicular to its length | ML²/3 |
| Solid sphere, about a diameter | 2MR²/5 |
| Thin hollow sphere (spherical shell), about a diameter | 2MR²/3 |
| Solid sphere, about a tangent | 7MR²/5 |
| Thin hollow sphere, about a tangent | 5MR²/3 |
| Solid cylinder, about its own axis | MR²/2 |
| Thin hollow cylinder, about its own axis | MR² |
| Solid cylinder, through the centre, perpendicular to its axis | M(R²/4 + L²/12) |
| Rectangular plate (sides a and b), through the centre, perpendicular to the plate | M(a² + b²)/12 |
| Annular disc (radii R₁ and R₂), through the centre, perpendicular to its plane | M(R₁² + R₂²)/2 |
| Solid cone, about its own axis | 3MR²/10 |
Questions students ask
What is the moment of inertia of a thin rod through the centre, perpendicular to its length?
I = ML²/12, so the radius of gyration is k = L/(2√3).
How is the moment of inertia of a thin rod through the centre, perpendicular to its length derived?
Take a piece dx at distance x from the centre. Its mass is dm = (M/L) dx. I = ∫ x² dm = (M/L) ∫ x² dx from −L/2 to L/2. = (M/L) × [x³/3] = (M/L) × (L³/24 + L³/24) = ML²/12.
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