Back to course
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 Moments
Edexcel A Level Maths: 4.4 Centres of Mass (Moments, Resultant Force)Curated by Kepler, not produced by us. Watch it before the notes below.
Open on YouTube

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
1

Non-uniform rods

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

∑M=0 ⇒ locate centre of mass\sum M = 0 \ \Rightarrow\ \text{locate centre of mass}
Take moments to find the unknown distance.
Example 1A non-uniform rod balances on a support 1.2 m from one end. What does this tell you?
1At the balance point, moments are equal either side.
2The centre of mass is at the support.
Answer
Centre 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\text{uniform: weight at centre}
Symmetric body.
∑M=0⇒xˉ\sum M = 0 \Rightarrow \bar{x}
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.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. A uniform rod’s weight acts at:

    • the centre
    • one end
    • the support
    • the heavier end
    Show answer and explanation

    Answer: the centre

    By symmetry.

  2. A non-uniform rod’s weight acts at:

    • an unknown point
    • the centre always
    • one end always
    • the support always
    Show answer and explanation

    Answer: an unknown point

    Found by moments.

  3. To locate the centre of mass you:

    • take moments (ΣM = 0)
    • resolve forces only
    • differentiate
    • measure the length
    Show answer and explanation

    Answer: take moments (ΣM = 0)

    Equilibrium of moments.

  4. If a rod balances on one support, its centre of mass is:

    • above the support
    • at one end
    • off the rod
    • at the lighter end
    Show answer and explanation

    Answer: above the support

    Moments balance there.

  5. The rod’s weight should be treated as:

    • a force at the centre of mass
    • ignored
    • spread evenly only if non-uniform
    • two forces
    Show answer and explanation

    Answer: a force at the centre of mass

    Single force.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 5 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.