Moments and Equilibrium of Bodies

Lessons

Senior 4 · Physics

Moments and Equilibrium of Bodies

4 lessons

Lesson 3 of 4

Centre of Gravity, Stability and Toppling

25 min

Rocks that should fall, and do not

In the Matobo hills of Zimbabwe, huge granite boulders sit stacked on top of one another, some on bases far narrower than themselves, and have stood that way for thousands of years. A minibus with sacks piled high on its roof, on the other hand, can roll over on a bend that an empty one takes easily. Both come down to one point in each body, its centre of gravity, and to where the vertical line through that point falls.

Centre of gravity and total weight

Every particle of a body is pulled down by gravity. In a uniform gravitational field, all these small weights add up to one total weight $W = m g$ that acts at a single point, the centre of gravity (G). For a uniform body with a centre of symmetry (a rod, a rectangle, a sphere) G is at that centre. G need not lie inside the material: the centre of gravity of a ring is at the empty centre.

Combining parts. The total weight must have the same moment about any point as all the separate weights together. Taking moments about an origin O for parts of weight $W_1, W_2, \ldots$ whose centres of gravity are at $x_1, x_2, \ldots$ :

(W_1 + W_2 + \cdots)\,\bar{x} = W_1x_1 + W_2x_2 + \cdots \quad\Rightarrow\quad \bar{x} = \frac{\sum W_i x_i}{\sum W_i}

and similarly for $\bar{y}$ . Because $W = m g$ , you may use masses (or, for a flat card of uniform thickness, areas) instead of weights.

Experiment 3.1: the centre of gravity of an irregular card

Apparatus: a piece of stiff card of irregular shape with three small holes near its edge, a pin held in a clamp, a plumb line (a thread with a small weight), a pencil.

Figure 3.1: The plumb-line method. A freely hanging body comes to rest with G vertically below the point of suspension.

Hang the card from the pin through one hole so that it swings freely. Hang the plumb line from the same pin and, when both are still, mark the line on the card.
Repeat from a second hole. The two lines cross at G.
Check with the third hole: its line should pass through the same point. Finally balance the card on a pencil point at G.

Why it works: a hanging body is in equilibrium only when its weight has no moment about the pin, that is when G lies on the vertical line through the pin.

Definition

The centre of gravity of a body is the point through which the whole weight of the body appears to act, whatever the body's orientation. For a system of parts, $\bar{x} = \frac{\sum W_i x_i}{\sum W_i}$ . The centre of mass is defined in the same way with masses; in a uniform field the two points coincide. Example: a uniform metre rule has its centre of gravity at the 50 cm mark.

Worked example 1: a rod with a load

A uniform rod 2.0 m long and weighing 20 N carries a 30 N weight at one end. Where is the centre of gravity of the whole?

\bar{x} = \frac{20 \times 1.0 + 30 \times 2.0}{20 + 30} = \frac{80}{50} = 1.6\,\mathrm{m} \text{ from the unloaded end}

The rod balances on a support 1.6 m from the unloaded end, closer to the heavy end.

Worked example 2: an L-shaped plate

A uniform L-shaped metal plate is made of a rectangle 40 cm × 10 cm lying along the $x$ axis from the origin, and a rectangle 10 cm × 30 cm standing on its left end (from $y = 10$ cm to $y = 40$ cm). Find G.

Areas and centres: $A_1 = 400\,\mathrm{cm^2}$ at (20, 5); $A_2 = 300\,\mathrm{cm^2}$ at (5, 25).

\bar{x} = \frac{400 \times 20 + 300 \times 5}{700} = 13.6\,\mathrm{cm}, \qquad \bar{y} = \frac{400 \times 5 + 300 \times 25}{700} = 13.6\,\mathrm{cm}

G is at (13.6 cm, 13.6 cm), which lies outside the metal, in the corner of the L.

Stable, unstable and neutral equilibrium

A body in equilibrium may react to a small push in three ways (Figure 3.2).

Figure 3.2: The three kinds of equilibrium, shown by a cone. The dashed line is the vertical through G.

Kind	After a small displacement…	Centre of gravity	Example
stable	the weight produces a moment that returns the body	rises when displaced	a cone on its base; a ball in a bowl
unstable	the weight produces a moment that moves it further	falls when displaced	a cone on its tip; a pencil on its point
neutral	the weight produces no moment; it stays in its new position	stays at the same height	a cone on its side; a ball on a level floor

In energy terms, a body in stable equilibrium sits at a minimum of gravitational potential energy, and one in unstable equilibrium at a maximum.

Toppling

Tilt a box about one bottom edge (Figure 3.3). As long as the vertical line through G falls inside the base, the weight's moment about the edge turns the box back. When the line passes outside the base, the moment turns the other way and the box topples. At the critical angle $\theta_c$ the line passes exactly through the edge; for a uniform box of base width $b$ with G at height $h$ ,

\tan\theta_c = \frac{b/2}{h}

Figure 3.3: Toppling happens when the vertical through G passes outside the base.

So a body is more stable when its centre of gravity is low and its base is wide. Racing cars are built low and wide; a loaded lorry should carry its heaviest goods at the bottom; a mug with a wide base is hard to knock over; you stand with your feet apart on a moving bus.

Worked example 3: how far can a bus lean?

A bus is 2.5 m wide with its centre of gravity 1.2 m above the road. (a) Through what angle can it be tilted sideways before it topples? (b) Passengers' luggage on the roof raises G to 1.6 m. Find the new angle.

\text{(a)}\;\; \tan\theta_c = \frac{1.25}{1.2} \;\Rightarrow\; \theta_c = 46^\circ \qquad \text{(b)}\;\; \tan\theta_c = \frac{1.25}{1.6} \;\Rightarrow\; \theta_c = 38^\circ

Raising the load reduces the margin of safety by 8°. On a bend, the sideways friction from the road adds a toppling moment, so a top-heavy vehicle overturns at a lower speed.

Warning

Common mistakes. (1) Thinking the centre of gravity must be inside the material. (2) Saying a body topples when G is "too high": it topples when the vertical through G passes outside the base; height only changes the angle at which that happens. (3) Adding positions without weighting them: $\bar{x}$ uses $W_{i} x_{i}$ , not just $x_{i}$ . Safety: heavy furniture such as bookshelves can topple onto children; fix tall shelves to the wall and keep heavy items on the lowest shelves.

Tip

For any toppling question, draw the vertical line through G. Inside the base: safe. Through the edge: critical. Outside: it falls.

Stable, unstable and neutral equilibrium (Flipping Physics). Watch how a ball, a bottle and a marker show the three kinds of equilibrium, and how the change in gravitational potential energy tells them apart.

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Practice

Define centre of gravity, and give an example of a body whose centre of gravity is not inside its material.
A uniform rod 1.0 m long weighs 5.0 N. A 10 N weight hangs 0.80 m from its left end. Find the centre of gravity of the system.
Masses of 2.0 kg, 3.0 kg and 5.0 kg lie on the $x$ axis at $x = 0$ , 1.0 m and 2.0 m. Find the centre of mass.
A uniform wardrobe is 0.60 m deep and 1.8 m tall. Find the angle through which it can be tilted forward before it topples.
State the kind of equilibrium of (a) a pencil balanced on its point, (b) a book lying flat on a table, (c) a football on a level field.
Explain why a double-decker bus can be tilted further without toppling when only its lower deck is loaded than when the same load is on its top deck.
A student says that a lorry carrying cement on its roof rack is as stable as one carrying the same cement on its floor, because "the weight is the same". Evaluate the claim.

Show answersHide answers

The point through which the whole weight appears to act. A ring, a hollow ball, an L-shaped plate or a boomerang.
$\bar{x} = \frac{5.0 \times 0.50 + 10 \times 0.80}{15} = 0.70\,\mathrm{m}$ from the left end.
$\bar{x} = \frac{2.0 \times 0 + 3.0 \times 1.0 + 5.0 \times 2.0}{10.0} = 1.3\,\mathrm{m}$ .
G is 0.90 m high and 0.30 m from the front edge: $\tan\theta_c = \frac{0.30}{0.90}$ , $\theta_c = 18^\circ$ . A tall, shallow wardrobe topples easily, which is why it should be fixed to the wall.
(a) unstable; (b) stable; (c) neutral.
Loading the lower deck keeps G low, so the vertical through G stays inside the wheels until a large tilt. The same load on the top deck raises G, which reduces $\theta_c = \tan^{-1}\frac{b/2}{h}$ , so the bus topples at a smaller angle.
The claim is wrong. The weight is the same, but on the roof it raises the centre of gravity. The critical tilt angle $\tan^{-1}\frac{b/2}{h}$ falls as $h$ rises, so the roof-loaded lorry topples at a smaller angle on a bend or a slope.

Quick check

A box stands on a level floor and is tilted about one edge. When does it topple?

Key points

The total weight acts at the centre of gravity; for combined parts $\bar{x} = \frac{\sum W_i x_i}{\sum W_i}$ . Find it for a card with a plumb line.
Stable: G rises when displaced; unstable: G falls; neutral: G stays level.
A body topples when the vertical through G passes outside its base; $\tan\theta_c = \frac{b/2}{h}$ . Low G and a wide base mean high stability.

Sources and credits

9.3: Stability, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Photo: Balancing Rocks in Matopos National Park.jpg by Susan Adams, Wikimedia Commons, licensed under CC BY-SA 2.0.
Photo: The Leaning Tower of Pisa SB.jpeg by Saffron Blaze, Wikimedia Commons, licensed under CC BY-SA 3.0.
Video: Stable, Unstable, and Neutral Equilibrium, Flipping Physics, embedded from YouTube.

The experiment, the three diagrams, the worked examples and the practice exercises are written by Ganzaa.

1Rocks that should fall, and do not

2Centre of gravity and total weight

Experiment 3.1: the centre of gravity of an irregular card

3Stable, unstable and neutral equilibrium

Toppling

Rocks that should fall, and do not

Centre of gravity and total weight

Stable, unstable and neutral equilibrium