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Statics · 9.1

Centre of Mass & Stability

The centre of mass is the weighted average position of a body's mass. Stability depends on where the weight's line of action meets the support base.

Explore the model
01

Find the balance point

Start with the physical model.

Treat each small mass as a contribution mᵢrᵢ. The centre of mass is the point whose position gives the same total first moment. In a nearly uniform gravitational field, the whole weight can be represented as acting through that point.

Simple definition
The centre of mass is the single point that represents the average position of an object's mass; stability depends on whether its weight line stays inside the support base.
Example
A hammer's centre of mass lies closer to its heavy head, and it tips when the vertical line through that point moves beyond its support edge.
Discrete massesxCM = Σmᵢxᵢ / Σmᵢ

The centre of mass is the weighted average position — each mass counts in proportion to how heavy it is.

Repeat independently for y and z

Continuous bodyrCM = (1/M)∫r dm

The same average for a solid object: add position × tiny mass over the whole body, then divide by total mass.

M = ∫dm

Stability testweight line lies inside base

An object stays upright while the vertical line from its centre of mass falls inside its footprint; past the edge, it tips.

At the edge: tipping threshold

02

Interactive statics laboratory

Change one condition and inspect the balance.

Every control reports its unit. Read the free-body model and the numerical check together.

Move two point masses across a massless 8 m platform and widen or narrow its support base.

Centre of mass-0.63 mΣmx ÷ Σm

Weight-line testInside baseStable against tipping

The vertical line through the centre of mass meets the interior of the support base.

03

Reason before calculating

Three ideas to keep visible.

01

Use symmetry first

A uniform object's centre of mass lies on every symmetry line or plane, often removing most of the calculation.

02

It need not lie in material

The centre of mass of a ring or curved object can lie in empty space.

03

Lower and wider is safer

A lower centre of mass and wider base require a larger tilt before the weight line reaches an edge.

04

Centre of gravity, and finding it

The centre of gravity is the single point through which the whole weight of a body can be taken to act. In the almost uniform field at the Earth's surface it sits where the centre of mass does, and exam questions use the term centre of gravity, so treat the two names as the same point unless a question says otherwise.

05

The plumb-line method

To find it for an irregular lamina: make a small hole near the edge of the card and hang it on a pin held horizontally, loose enough to swing freely. Hang a plumb line from the same pin and, once both have stopped moving, draw the line the string marks across the card. The centre of gravity lies somewhere on that line, because the card only rests when its weight acts straight down through the point of support. Repeat from a second hole well away from the first: the centre of gravity is where the two lines cross. A third hole gives a check — all three should meet at one point, and a small triangle between them means a line was drawn carelessly.

04 · Quick check

A rigid object is slowly tilted. When does it reach the ideal tipping threshold?

Choose an answer to reveal the reasoning.

EasyMasses of 3.0 kg and 1.0 kg sit at the ends of a light 2.0 m rod. How far from the 3.0 kg mass is the centre of mass?
  1. x(CM) = (3.0 × 0 + 1.0 × 2.0) ÷ 4.0.
  2. x = 0.50 m from the heavier end.

Result0.50 m from the 3.0 kg mass

MediumMasses of 2.0 kg and 6.0 kg are at x = 0 m and x = 4.0 m. Locate their centre of mass.
  1. Choose the stated origin and keep both coordinates signed.
  2. xCM = (2.0×0 + 6.0×4.0)/(2.0 + 6.0).
  3. The larger mass pulls the weighted average closer to x = 4.0 m.

ResultxCM = 3.0 m

HardA uniform L-shaped plate is made of two 1.0 m × 0.20 m strips of the same sheet joined at right angles. Locate its centre of mass relative to the corner.
  1. Each strip's CM sits at its centre: (0.50, 0.10) and (0.10, 0.50) m; equal masses.
  2. x(CM) = (0.50 + 0.10)/2 = 0.30 m; y(CM) = (0.10 + 0.50)/2 = 0.30 m.
  3. The CM at (0.30, 0.30) lies outside the material of the L — perfectly allowed.

Result(0.30 m, 0.30 m) from the corner — off the plate itself

ChallengingA uniform box 0.40 m wide and 0.90 m tall stands on a lorry bed. At what sideways tilt angle does it topple, and how does loading the bottom half with extra mass change the answer?
  1. Topple when the weight line leaves the base edge: tanθ = (width/2)/(height of CM) = 0.20 ÷ 0.45.
  2. θ = tan⁻¹(0.444) ≈ 24°.
  3. Bottom-loading lowers the CM (say to 0.30 m): tanθ = 0.20/0.30 → θ ≈ 34° — lower CM, wider safety margin.

Result≈ 24° unladen; lowering the CM to 0.30 m raises it to ≈ 34°