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Centre-of-gravity investigation

Key idea: Hang a flat shape freely from different points and trace the plumb line each time. The lines cross at its centre of gravity, which lies below each suspension point when it rests.

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  • SEC G3 Combined Science Physics component 2027
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The centre of gravity of a rectangular lamina with uniform thickness and density is at its middle, but for an irregular flat shape there is no middle to measure. Instead, hang the shape up and let gravity show you: a plumb line marks a vertical line through the centre of gravity, and two or three such lines cross at it. Use a thin, stiff card, a firmly clamped support and a pin that lets the card swing freely.

Find the centre of gravity of a plane lamina

When a flat shape hangs freely from a point and stops swinging, its centre of gravity is directly below the point of suspension. Its weight then has no moment about that point. So a vertical line drawn down from the suspension point passes through the centre of gravity. Two lines from different suspension points locate it; a third line checks the result. The card changes orientation each time, but the centre of gravity stays at the same position on the card.

One suspension gives a line; several suspensions locate the centre of gravityTwo panels show the same uniform card. On the left it hangs freely from hole A, with a plumb line from the same pin. Two marked points trace the vertical line on the card. On the right the card is laid flat after separate suspensions from holes A, B and C. Their three traced lines cross at G, the centre of gravity. The ideal construction has an exact crossing; real drawn lines can form a small region. The card's orientation changes during suspension, while the centre of gravity stays fixed on the card.1. Trace one vertical line2. Repeat from other holessame pin at Amark two pointsplumb lineABCGG stays fixed on the card.The card must swing freely.A third trace checks the crossing.
One suspension gives a line; several suspensions locate the centre of gravityTwo panels show the same uniform card. On the left it hangs freely from hole A, with a plumb line from the same pin. Two marked points trace the vertical line on the card. On the right the card is laid flat after separate suspensions from holes A, B and C. Their three traced lines cross at G, the centre of gravity. The ideal construction has an exact crossing; real drawn lines can form a small region. The card's orientation changes during suspension, while the centre of gravity stays fixed on the card.1. Trace one vertical linesame pin at Amark two pointsplumb lineABCGG stays fixed on the card.A third trace checks the crossing.2. Repeat from other holesThe card must swing freely.
Trace a vertical plumb line from each suspension hole on the same card. Ideal traces meet at G; real traces may form a small region.
  1. Make three small holes near the edge of the card, on different sides of the shape. Choose positions that will give lines crossing at a clear angle.
  2. Hang the card from a pin through one hole, clamped in a stand. Check that the card swings freely.
  3. Hang a plumb line (a thread with a small mass) from the same pin. When both are still, mark two points on the card directly behind the thread, far apart, and join them with a ruler.
  4. Repeat from the other two holes.
  5. The lines should cross at one point, or form a small region because of drawing and reading errors. For this solid card, gently balancing it on a rounded pencil tip near the crossing is an additional check. A shape with a gap can have its centre of gravity in the gap; the plumb-line method still works.

Check your understanding 1 Explain the freely hanging equilibrium

Why does the centre of gravity of a freely hanging card lie on the plumb line from its suspension point?
Show answer

When the card hangs freely and at rest, its weight’s line of action passes through the suspension point, giving zero moment about it. Its centre of gravity is directly below the point and therefore lies on that vertical plumb line. The line does not necessarily pass through the card’s geometric middle; free swinging and a common suspension point are essential.

Guided practice 1

When the three lines do not meet at one point

About 4 min

Problem

The three traced lines form a small triangle. Measured from the bottom-left corner of the card, the corners of the triangle are at (6.2 cm, 4.1 cm), (6.5 cm, 4.3 cm) and (6.3 cm, 4.4 cm). Use the centre of the triangle as one reasonable estimate of the position. Round each coordinate to 0.1 cm.

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Show the steps Optional working
Unit: cm
Unit: cm
Show the steps Optional working
Unit: cm
Unit: cm

Hints

Hint 1

estimate a position

Take the centre of the triangle: the mean of the three corners in each direction.

Hint 2

describe the spread

How far do the corners spread from that centre?

Show solution step by step

  1. Estimate the position

    Method

    Average the corners.

    Reason

    The centre of the small triangle gives one practical estimate. The intersections share lines, so averaging them does not give three independent measurements.

    Working

    Answer

    x = (6.2 + 6.5 + 6.3) ÷ 3 = 6.33 ≈ 6.3 cm

    ,

    y = (4.1 + 4.3 + 4.4) ÷ 3 = 4.27 ≈ 4.3 cm

  2. Describe the spread

    Method

    In each coordinate, the corners are within 0.2 cm of the rounded estimate.

    Reason

    The small spread indicates disagreement between the drawn lines. It does not include every possible error, such as friction at the pin.

    Working

    Answer

    Estimate: (6.3 cm, 4.3 cm) from the stated corner, with a coordinate spread of about 0.2 cm. Friction at the pin could shift all the lines, even if they cross close together.

  3. State the estimate

    Working

    Answer

    The estimated position is (6.3 cm, 4.3 cm). Each coordinate of the intersections is within 0.2 cm of that rounded estimate; this describes their spread, not every source of error.

Check your understanding 2 What does the small triangle tell you?

How far are the intersection coordinates from the rounded estimate (6.3 cm, 4.3 cm)? Explain why their small spread does not account for every error in the centre-of-gravity position.

Show answer

The x coordinates differ from 6.3 cm by −0.1, +0.2 and 0 cm. The y coordinates differ from 4.3 cm by −0.2, 0 and +0.1 cm. Each coordinate is within 0.2 cm of the estimate. The three intersections share the traced lines, so they are not three independent measurements. Friction at the pin or the same sighting error on each trace could shift the lines together; a small triangle alone would not reveal that error. Check the method and repeat the suspensions after correcting any problem.

Limitations and improvements

SourceEffectImprovement
The pin fits the hole tightly, so the card cannot swing freelyThe card stops before its centre of gravity is below the pin, so the line misses itUse a slightly larger hole and a smooth pin; tap the card and let it settle.
Marking the thread’s position by eye from the sideThe drawn line is not where the thread isLook straight at the card, mark two points far apart directly behind the thread and join them with a ruler.
Suspension holes close togetherThe lines cross at a shallow angle, so a small drawing error moves the crossing a long wayChoose holes on different sides so the traced lines cross at a clear angle.
Thread or card still swingingThe line is drawn at an angleWait until both are completely still.

Check your understanding 3 Adding a mass to the card

Plan an extension to find how the mass of plasticine added at one fixed corner of a card affects its centre of gravity. State what you would change, measure and keep the same, and the result you expect. On paper, sketch the card with its bare centre of gravity and the fixed added-mass position, and mark the expected direction of the new centre-of-gravity shift. Explain why a repeat trace from one suspension hole alone would not locate the new point.

Show answer

First locate the bare card’s centre of gravity with three plumb lines. Keep the same card and fix plasticine at the same corner each time. Change the plasticine mass and find the new centre of gravity with three plumb lines. Repeat with lumps of different mass, measuring each one on a balance. Plot the distance the centre of gravity moves against the mass of plasticine. The centre of gravity should move towards the added mass, further for a larger mass. Self-review the sketch: mark the bare centre of gravity, the fixed corner carrying the added mass and a shift towards that corner. Do not assume the shift is proportional to added mass. Each suspension gives only one line of possible positions, so use different holes that give clearly crossing traces after each mass change, and check their spread. Keep the plasticine firmly attached without changing its chosen position or allowing it to distort the card; handling and measurement still require supervised apparatus.

Common mistakes

  • Hanging the plumb line from a different point from the card.
  • Marking the thread before the card and plumb line have stopped swinging.
  • Sighting the thread at an angle instead of looking straight at the card.
  • Using suspension holes close together, so the lines cross at a shallow angle.
  • Assuming the centre of gravity is at the geometric middle, or always inside the material of the shape.

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Syllabus and review details

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