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UY1: Centre Of Mass Of A Cone

Derive the centre of mass of a uniform solid cone using disk integration and geometric similarity.

Reading settings
  • University Physics Year 1
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Cluster role + pathway links

This page is the archived worked-example derivation in the UY1 mechanics pathway.

1) At a glance

  • Prerequisite: CM integral, volume elements, similar triangles.
  • Outcomes: derive CM location of a uniform solid cone.
  • Key result:
    z_CM = 3h/4
    measured from the apex along the symmetry axis (equivalently h/4 from the base).
Common traps (reference point)
  • Always state the reference point: here z = 0 at the apex and z = h at the base. - Don’t confuse the slice radius r(z) = (R/h)z with the cone’s base radius R.

2) Setup (model, assumptions, coordinates/signs)

A solid cone with its apex at the origin and axis along z. The base has radius R at height h. A thin disc of radius r, thickness dz and mass dm sits at height z. The centre of mass is marked on the z-axis.
Each thin disc at height z has radius r = (R/h)z. Summing the discs puts the centre of mass on the axis at z = 3h/4 from the apex.
  • Uniform solid cone of height h, base radius R, constant density ρ.
  • Place apex at z = 0 and base at z = h.
  • By symmetry:
    x_CM = 0, y_CM = 0
    So only z_CM is needed.
  • Radius at height z from similar triangles:
    r(z) = (R/h)z

3) Core method / derivation

z_CM = 1/M∫ z dm

Take a thin disk of thickness dz at height z:

dm = ρ (π r²) dz = ρπ(R²/h²)z² dz

Total mass:

M = ∫₀^h dm = ρπR²/h²∫₀^h z² dz = ρπ R²h/3
z_CM = 1/M∫₀^h z dm; = (1/M)ρπR²/h²∫₀^h z³ dz; = (1/M)ρπ(R²/h²)h⁴/4; = 3h/4

Checks:

  • Units: z_CM has units of length.
  • Sign: z_CM > 0 along the chosen positive axis.
  • Limits: for fixed h, changing ρ does not change CM location.

Intuition:

  • Slices near the base have much larger area (∝ z²), so they contribute more mass and pull the CM toward the base.

4) Worked example

Worked example 1

Locate the centre of mass of a cone

Problem

A uniform solid cone has height h = 20 cm. Where is its centre of mass?

Show full solution

  1. Use the derived result

    Method

    Apply z_CM = (3/4)h, measured from the apex.

    Reason

    Each disc slice has mass proportional to z², so the mass is concentrated towards the base.

    Working

    Answer

    z_CM = (3/4)(20 cm) = 15 cm.

  2. State the answer

    Working

    Answer

    The centre of mass is on the axis, 15 cm from the apex, which is 5 cm above the base.

5) Practice set

Check your understanding 1

A uniform solid cone has h = 0.80 m. How far above the base is its centre of mass?

Show hint

First find 3h/4 from the apex.

Show answer

From the apex, z_CM = (3/4)(0.80) = 0.60 m, so it is 0.80-0.60 = 0.20 m above the base: a quarter of the height.

Check your understanding 2

The base radius of a uniform cone doubles while its height stays fixed. Does z_CM change?

Show hint

Look at the final formula: which dimensions appear in it?

Show answer

No. Every slice’s mass scales by the same factor, 4, so the weighting along z is unchanged and z_CM = (3/4)h depends only on h.

Check your understanding 3

True or false: a thin hollow conical shell has the same centre of mass as a solid cone of the same height.

Show hint

How does the mass of each slice grow with z for a shell compared with a solid?

Show answer

False. For the shell, the mass of each ring grows only in proportion to z, not z², so the mass is less concentrated towards the base. Its centre of mass is at (2/3)h from the apex, not (3/4)h.

6) Summary + next steps

  • Symmetry reduces the problem to one coordinate.
  • Disk slicing with r(z) = (R/h)z gives z_CM = 3h/4 from apex.
  • Always report reference point (apex or base) explicitly.

Next steps:

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Syllabus and review details

No official syllabus alignment is listed for this lesson.

Last reviewed:

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