Describe, qualitatively, the effect of the position of the centre of gravity on the stability of simple objects

1.5.3 Centre of Gravity

Objective (AO1)

Describe, qualitatively, how the position of the centre of gravity (CG) – both vertically and horizontally – influences the stability of simple objects.

Key Definitions (AO1)

• Centre of gravity (CG): the point at which the total weight of an object can be considered to act.
• Stability (AO1): an object is stable when the vertical line through its CG falls inside the base of support; it is conditionally stable when the line passes exactly through an edge, and unstable when the line falls outside the base.
• Base of support: the area of contact between the object and the surface on which it rests.

How CG Position Affects Stability (AO1)

Vertical position (height of CG)

• A low CG (small height above the base) shortens the lever arm when the object is tilted, producing a smaller overturning torque and therefore greater stability.
• A high CG (large height) lengthens the lever arm, so even a small tilt creates a large overturning torque, making the object less stable or unstable.

Horizontal position (offset of CG)

• If the CG is centred over the base, the vertical line through the CG lies well inside the base – the object is stable.
• If the CG is shifted towards an edge, the line may be close to or outside the edge; the object can become conditionally stable or unstable even when the CG is low.

Restoring Torque (Qualitative)

When an object is tilted about a point on its edge (the pivot), the weight \(W\) acting through the CG produces an overturning torque

\[

\tau = W \times d

\]

• \(W\) – weight of the object (N).
• \(d\) – horizontal distance from the CG to the pivot. For a small tilt, \(d \approx h\sin\theta\) where \(h\) is the vertical height of the CG and \(\theta\) the tilt angle.

Thus a larger \(h\) gives a larger \(d\) for the same \(\theta\), increasing the torque that tends to tip the object.

AO2 – Predicting Stability (Decision‑Tree)

1. Draw the vertical line from the CG to the supporting surface.
2. Locate where this line meets the base of support.

• Inside the base → stable.
• Exactly on an edge → conditionally stable (any further disturbance will cause tipping).
• Outside the base → unstable.

Worked Example (Rectangular Block)

Consider a rectangular block 200 mm × 100 mm × 50 mm placed on a flat table.

1. Locate the CG: for a uniform block the CG is at the geometric centre – 100 mm from each side and 25 mm above the table.
2. Draw the vertical line: it passes through the centre of the 200 mm × 100 mm face, i.e. 50 mm from each edge.
3. Apply the decision‑tree: the line meets the base well inside the rectangle → the block is stable.
4. What if the block is turned on its thin edge? The base becomes 50 mm wide, the CG is still 25 mm above the table but now only 25 mm from each edge. The vertical line still lies inside, so the block remains stable, but the smaller base means a smaller disturbance will bring the line to the edge → the object is now only conditionally stable.

AO3 – Practical Determination of the CG of an Irregular Plane Lamina

String‑Suspension Method (Uniform Density Assumption)

• Apparatus: irregular lamina (e.g. cardboard cut‑out), two thin strings, small hook or pin, ruler, flat bench, fine‑point marker.
• Assumption: the lamina has uniform density; the CG coincides with the centre of mass.
• Procedure:

  1. Attach a string to a point near the edge of the lamina and suspend it from a fixed support so the lamina can rotate freely.
  2. Allow the lamina to come to rest. The string now passes through the CG.
  3. Without moving the lamina, draw a straight line on it in the direction of the string (the “plumb line”).
  4. Repeat the suspension from a different point on the lamina and draw a second plumb line.
  5. The intersection of the two lines marks the CG.

• Safety / Practical notes:

  • No special hazards – ensure the bench is stable and the strings are not under excessive tension.
  • Use a fine‑point marker so the drawn lines do not affect the lamina’s balance.

Alternative Method – Balancing on a Point

• Place the lamina on a sharp edge (e.g. the tip of a needle) and slowly move the edge until the lamina balances without rotating.
• The point of contact lies directly beneath the CG; tracing this point on the lamina gives the CG location.
• This method also assumes uniform density and is useful when strings are unavailable.

Examples of Simple Objects

Design Tips to Improve Stability (AO2)

• Lower the CG: add weight at the base (e.g. sandbags on a crane, ballast in a boat).
• Widen the base of support: broader legs on a table, outriggers on a camera tripod.
• Shift the CG horizontally: centre the load over the base (e.g. place items symmetrically on a shelf).
• Use shapes with naturally low CGs: pyramids with a wide base, conical roofs.

Common Misconceptions (AO1)

• “A heavier object is always more stable.” – Stability depends on where the weight acts, not on the magnitude of the weight.
• “If an object does not move, its CG must be at the centre of the base.” – The CG can be anywhere within the base; as long as the vertical line through the CG stays inside, the object remains in equilibrium.
• “Only the height of the CG matters.” – Both vertical height and horizontal offset relative to the base are crucial.

Summary Checklist (AO1 & AO2)

1. Locate the CG (experiment, diagram or given data).
2. Determine the vertical height of the CG above the supporting surface.
3. Draw the vertical line from the CG to the base of support.
4. Apply the decision‑tree:

   • Inside the base → stable.
   • On the edge → conditionally stable.
   • Outside the base → unstable.

5. Predict the effect of a small disturbance and suggest practical ways to increase stability if required.