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Moment of inertia of a rod
Intermediate
About the centre or the end of a uniform rod.
I = mL²/12; mL²/3About its centre I = mL²/12; about an end I = mL²/3 (parallel axis theorem with d = L/2).
1.2 kg, 1 m: 0.1 and 0.4 kg·m².
Use the right axis.
In more detail
For a thin uniform rod of mass M and length L rotating about its centre, I = ML²/12. About one end, I = ML²/3.
Real-world uses
- Swinging doors and metre rulers use this moment of inertia in their dynamics.
- Robotic arms are modelled as rods to estimate joint torques.
- A metre rule balanced on a fingertip has a moment of inertia that is smaller about its centre than about its end.
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Frequently asked questions
Why is the end value bigger?
Because more mass is further from the axis, so the moment of inertia is larger.
Does the rod's thickness matter?
For a thin rod, no. For a thick one, the formula changes.
Why is the moment of inertia larger about the end?
More of the mass is further from that axis, and the formula uses the square of each distance from the axis.
Related topics in Rotational Motion
Centre of gravity · Stability and toppling · Moment of inertia · Moment of inertia of a ring · Moment of inertia of a disc · Moment of inertia of a solid sphere