4.4MechanicsStretch

Centres of Mass

A non-uniform rod has its weight acting at an unknown point. By taking moments about a support, you can locate this centre of mass from the equilibrium conditions.

24 min Video by Zeeshan Zamurred Moments

Edexcel A Level Maths: 4.4 Centres of Mass (Moments, Resultant Force)Watch the full walkthrough before the notes below.

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What you'll be able to do

Understand centre of mass for non-uniform rods
Take moments to locate it
Combine with equilibrium equations
Interpret the result

Non-uniform rods

For a $uniform$ rod the weight acts at the centre. For a $non-uniform$ rod the weight acts at an unknown distance from one end — this is what you solve for.

\sum M = 0 \Rightarrow locate centre of mass

Take moments to find the unknown distance.

1At the balance point, moments are equal either side.

2The centre of mass is at the support.

AnswerCentre of mass is at the balance point (1.2 m).

Tip — If a rod balances on a single support, its centre of mass is directly over that support.

Formula recap

uniform: weight at centre

Symmetric body.

\sum M = 0 \Rightarrow \overset{x}{ˉ}

Locate the centre of mass.

Common mistakes to avoid

Assuming a non-uniform rod’s weight acts at the centre.

Its weight acts at an unknown point you must find.

Ignoring the rod’s weight entirely.

Include the weight as a force at the centre of mass.

Key takeaways

Uniform rod: weight at the centre.
Non-uniform rod: centre of mass is unknown — find it by moments.
Balancing on a single support ⇒ centre of mass is above it.

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