Describe an experiment to determine the position of the centre of gravity of an irregularly shaped plane lamina

1.5.3 Centre of Gravity

Learning Objective (AO1)

Describe an experiment to determine the position of the centre of gravity of an irregularly shaped plane lamina.

Key Statement (Syllabus wording)

The centre of gravity (CG) is the point through which the resultant weight of an object acts.

What you will be asked to do (exam checklist)

• Identify two different equilibrium lines by supporting the lamina at two points.
• Draw the two lines on the lamina (or on a transparent sheet).
• Locate the intersection of the two lines – this point is the centre of gravity.

Core Definition

The centre of gravity (CG) is the point through which the resultant weight of an object acts.

Supplementary (Extended) Note

For a uniform lamina the CG coincides with the centre of mass because the density is the same at every point. If the lamina is divided into small elements of mass m_i at positions (x_i, y_i), the coordinates of the centre of gravity are

\(xG = \dfrac{\sum mi xi}{\sum mi},\qquad

yG = \dfrac{\sum mi yi}{\sum mi}\)

This relation is useful for theoretical work or for an extension activity, but it is not required for the practical experiment.

Example (optional extension)

Suppose a rectangular cardboard (10 cm × 6 cm) is cut into three equal strips, each of mass 2 g, 3 g and 5 g, placed side‑by‑side. Using the formula above gives

\(x_G = \dfrac{2(2)+3(5)+5(8)}{2+3+5}=5.6\;\text{cm}\) from the left edge.

Aim

To locate the centre of gravity of an irregularly shaped thin lamina by using the principle that the lamina balances when supported at its CG.

Apparatus

Experimental Principle (static equilibrium)

• When a lamina is supported at two points, the line joining those points is a line of action of the resultant weight.
• At equilibrium the net moment about any point on that line is zero; therefore the line must pass through the centre of gravity.
• Two different support pairs give two equilibrium lines; their intersection is the unique CG.

Procedure (step‑by‑step)

1. Set‑up the supports. Secure the mounting board on a stable table. Fix the two thin supports so that they are parallel and spaced about 10 cm apart. Ensure they can slide or be repositioned without wobble.
2. Place the lamina. Lay the irregular lamina on the supports so that it can rotate freely. Use a plumb line to check that the board is level.
3. Find the first equilibrium line (L₁).

   • Gently lower the lamina until it just rests on the two support points without tipping to either side.
   • Mark the two contact points on the lamina (or on a transparent sheet placed over it) with a pencil.
   • Using a ruler, draw a straight line through the two marks and label it L₁.

4. Obtain a second equilibrium line (L₂).

   • Without moving the board, rotate the lamina to a different orientation (or move the supports to a new pair of points) and repeat step 3.
   • Mark the new contact points and draw the second line, label it L₂.

5. Locate the centre of gravity. Extend L₁ and L₂ until they intersect. Mark the intersection as point G. This is the centre of gravity of the lamina.
6. Record coordinates. Choose a convenient reference corner (e.g., the lower‑left corner). Measure the horizontal distance xG and vertical distance yG from that corner to point G. Record the values with appropriate significant figures.
7. Verification (optional). Support the lamina directly at the measured point G. It should remain horizontal and show no tendency to rotate.

Data Recording Table

Analysis

• Each equilibrium line must pass through the centre of gravity; therefore the intersection of any two lines gives the exact CG.
• No further calculations are required unless you wish to compare the measured coordinates with a theoretical prediction (using the mass‑weighted formula).
• Uncertainty estimate. If the ruler has a least count of 0.5 cm, the uncertainty in each coordinate can be taken as ±0.5 cm. Propagate this to give an overall uncertainty for the position of G.
• Confirm the result by supporting the lamina at the recorded point G – it should balance without rotation.

Safety and Precautions

• Secure the mounting board to prevent accidental movement.
• Use a light plumb‑line weight to avoid damaging the lamina.
• Do not force the lamina into equilibrium; allow it to settle naturally.
• Handle sharp edges of the cardboard or supports with care.

Extension / Evaluation Questions

• How would the method be altered if the lamina were non‑uniform (density varies across the surface)?
• Explain why the centre of gravity of a symmetric shape such as a rectangle lies at its geometric centre.
• Identify possible sources of experimental error (e.g., parallax when marking points, ruler precision, movement of supports) and suggest ways to minimise them.
• Estimate the uncertainty in the measured coordinates of G based on the ruler’s least count and discuss how this affects the final result.