All subjects · Mathematics · WME03 Mechanics 3

Centres of mass of composite bodies

Edexcel IAL Mathematics · WME03 Mechanics 3

Real objects are often made of standard shapes joined together, or with a piece cut out. Use a table of masses and distances, exactly as in M2, but with volumes and the M3 standard results.

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The table method

1. Choose an origin and axis (often the common plane face).

2. List each part with its mass (for a uniform solid, mass is proportional to volume) and the distance of its centre of mass.

3. Σ(m x) = (Σm) x̄.

A removed part has negative mass in the table.

Signs

Measure all distances from one point along one axis, with a clear positive direction. A hemisphere and a cone joined at their plane faces have their centres on opposite sides of the join.

Typical questions

"Toys" made from a hemisphere and a cone, a cylinder with a hemisphere or cone removed, a lamina with a hole. A common follow-up asks when the centre of mass lies on the join, which decides whether the toy can rest in any position on its curved surface.

Worked example

A uniform solid hemisphere of radius 3 cm and a uniform solid cone of radius 3 cm and height 8 cm, made of the same material, are joined at their plane faces. Find the distance of the centre of mass from the common face.

  1. Volumes: hemisphere (2/3)π(27) = 18π; cone (1/3)π(9)(8) = 24π.
  2. Distances from the face (cone side positive): cone +2, hemisphere -9/8.
  3. x̄ = (24π × 2 - 18π × 9/8)/(42π) = 27.75/42.

Answer: 0.661 cm from the common face, inside the cone

Key idea: Mass ∝ volume; table of parts; removed parts are negative; Σmx = (Σm)x̄.

Check yourself

  1. In the worked example, find the distance of the centre of mass from the common face, to 3 significant figures.
    Show answer

    0.661 cm. x̄ = 0.661 cm.

  2. A uniform solid cylinder has radius 2 cm and height 6 cm. A hemisphere of radius 2 cm is removed from one end, leaving a hollow with its plane face level with that end. Find the distance of the centre of mass of the remaining solid from the other (intact) end, to 3 significant figures.
    Show answer

    2.36 cm. Cylinder: 24π at 3. Hemisphere: (16/3)π at 6 - 0.75 = 5.25. x̄ = (72 - 28)/(24 - 16/3) = 2.36 cm.

  3. A toy is a uniform solid hemisphere of radius 2 cm joined at its plane face to a uniform solid cone of the same radius and material. Find the height of the cone for which the centre of mass lies on the common face, to 3 significant figures.
    Show answer

    3.46 cm. Moments about the face: (1/3)πr2h × h/4 = (2/3)πr3 × 3r/8, so h2 = 3r2 and h = r sqrt 3 = 3.46 cm.

  4. A uniform square lamina OABC has side 4 cm, with O at the origin and sides along the axes. A circular hole of radius 1 cm centred at (1, 1) is cut out. Find x̄ for the remaining lamina, to 3 significant figures.
    Show answer

    2.24 cm. Square: area 16 at x = 2. Hole: area π at x = 1. x̄ = (32 - π)/(16 - π) = 2.24 cm.

  5. How is a removed part shown in the table for a composite body?
    • With a negative mass
    • With a negative distance
    • With zero mass
    • It is left out
    Show answer

    With a negative mass. The original body equals the remainder plus the removed part, so the removed part is subtracted.

  6. Two uniform solids are made of the same material. What is each one's mass proportional to?
    • Its surface area
    • Its height
    • Its volume
    • Its radius
    Show answer

    Its volume. Same density, so mass = density × volume.

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Last updated 8 November 2026. Spotted a mistake? Tell us.