Centres of Mass summary

Key learning points

• Uniform shapes: centroid from symmetry.
• $\bar{x}=\frac{\sum m_i x_i}{\sum m_i}$ (similarly for $y$).
• Removed regions subtract their moment contribution.
• Toppling: track whether the vertical through the CoM stays inside the base.

Worked example 1 — Two particles

Masses $2\,\mathrm{kg}$ at $x=0$ and $4\,\mathrm{kg}$ at $x=6$. CoM?

Working

Mass-weighted average: $\frac{24}{6}=4$.

Worked example 2 — Uniform shapes: centroid from symmetry

Use this method from the Centres of Mass sheet: Uniform shapes: centroid from symmetry.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 3 — (similarly for )

Use this method from the Centres of Mass sheet: $\bar{x}=\frac{\sum m_i x_i}{\sum m_i}$ (similarly for $y$).

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 4 — Removed regions subtract their moment contribution

Use this method from the Centres of Mass sheet: Removed regions subtract their moment contribution.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 5 — Toppling: track whether the vertical through

Use this method from the Centres of Mass sheet: Toppling: track whether the vertical through the CoM stays inside the base.

Working

Each key learning point matches a question type on the printable worksheet.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.