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Centre of mass of shapes
Intermediate
Find by symmetry or integration.
Symmetry helps: a uniform rod's centre of mass is its midpoint, a triangle's is at one third of its height, a semicircle's disk at 4R/3π.
An L-shape: split into rectangles and use weighted averages.
Composite shapes with holes: subtract the hole.
In more detail
The centre of mass of a symmetric shape lies at its geometric centre. For more complex shapes, it is found by averaging position weighted by mass, using integration or symmetry.
Real-world uses
- Designers position the centre of mass of vehicles and aircraft carefully for stability.
- Balance boards and rulers are set to balance at their centre of mass.
- A structural engineer finds the centre of mass of a complicated beam shape by splitting it into simple parts.
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Frequently asked questions
How do I find the centre of mass of an L-shape?
Split it into rectangles, find each centre, and take the mass-weighted average.
Does a uniform rod's centre of mass lie at the middle?
Yes, for a uniform rod, the centre of mass is at its midpoint.
What is the quickest way to find a centre of mass?
Use symmetry first. Any symmetry axis or plane contains the centre of mass, which reduces the problem to one or two coordinates.
Related topics in Rotational Motion
Torque · Lever arm and moment · Equilibrium of a rigid body · Centre of gravity · Stability and toppling · Moment of inertia