Physics · Mechanics · Grade 10

Centre of gravity and equilibrium

How far can books be stacked over the edge of a table, and when does a box tip over?

The situation

In the library, try stacking identical books like a staircase at the edge of a table: the top book can stick right out beyond the table and “hang” in the air. The Leaning Tower of Pisa has stood for centuries although it leans by about 4°. And a tall box on a tilted board sometimes topples over without sliding. When does a body stay in equilibrium and when does it fall — and can this be calculated in advance?

Topic

Statics and equilibrium. Tower and bridge: where is the centre of gravity

Key ideas

Centre of gravity
Condition for equilibrium
Harmonic series

Formulas

x_{c} = \frac{\sum m_{i} x_{i}}{\sum m_{i}}

The centre of gravity is the point where the resultant of the gravitational forces acts.

x_{left} \leq x_{c} \leq x_{right}

Condition for standing: the centre of gravity of the part above each level must lie over the supporting area.

d_{k} = \frac{L}{2 k}, s_{N} = \frac{L}{2} (1 + \frac{1}{2} + \dots + \frac{1}{N})

Books: the shift of the $k$ -th book from the top and the greatest overhang (harmonic series).

\tan θ > \frac{b}{h} (topples), \tan θ > μ (slides)

A box on a tilted board ( $b$ is the width of the base, $h$ the height): whichever condition is met first is what happens.

\sum {\vec{F}}_{i} = 0, \sum M_{i} = 0

Conditions for the equilibrium of a rigid body. For a body with a fixed axis the second is enough — the reaction of the axle balances the forces.

Theory

Centre of gravity

The gravitational force acts on every part of a body — this is a system of parallel forces. Their resultant equals the total weight and passes through the centre of gravity (in a uniform field it coincides with the centre of mass (inertia)). If the vertical line drawn down from the centre of gravity falls inside the supporting area, the support reaction balances the moment. If the line falls outside the support, the gravitational force produces a turning moment about the edge of the support. A way to find the centre of gravity: hang the body from two points in turn and draw the plumb lines — they cross at the centre of gravity (“Disc” mode).

Books and the harmonic series

The top book can stick out $L / 2$ beyond the one below it, the top two together can stick out $L / 4$ beyond the third, the top three $L / 6$ beyond the fourth — each time the centre of gravity of all the books above lies exactly at the edge of the book beneath. In total: $\frac{L}{2} (1 + \frac{1}{2} + \frac{1}{3} + \dots)$ . With 4 books this is $\approx 1.04 L$ — the top book comes completely off the table! The harmonic series grows without limit, but very slowly.

Tilted board: toppling or sliding

As the tilt increases, the vertical line from the centre of gravity moves towards the lower edge of the box's base. At $\tan θ = b / h$ it reaches the edge — the box topples. Sliding begins at $\tan θ = μ$ . Whichever condition is met first is what happens: a tall, narrow box topples, a low, wide box slides. Look at the graph to see which line the $\tan θ$ curve crosses first.

Statics: system of forces and conditions of equilibrium

Statics finds out when the forces acting on a body balance one another, and determines unknown reaction forces and the centre of gravity. A force is specified by its magnitude, direction and point of application (line of action); several forces form a system of forces, which can be replaced by a single resultant. The resultant of parallel forces acting in the same direction equals $F_{1} + F_{2}$ and divides the distance between them into parts inversely proportional to the forces ( $F_{1} d_{1} = F_{2} d_{2}$ ); for antiparallel forces it equals $| F_{1} - F_{2} |$ and lies outside them, on the side of the larger force; equal antiparallel forces form a couple, which has only a moment (“Simple machines: lever, pulley” experiment). For a body with no axis of rotation to be in equilibrium we need $\sum \vec{F} = 0$ and $\sum M = 0$ about any point; for a body with a fixed axis, $\sum M = 0$ is enough. Lab work in Disc mode: hang loads and check that $M_{1} + M_{2} + M_{d i s c} = 0$ . Buildings, bridges and cranes are calculated using these conditions: the reaction at every support and the moment in every beam are found in advance.

Types of equilibrium and potential energy

If the centre of gravity of a disc on an axle is below the axle, after a small push the disc swings back — stable equilibrium; if it is above, the disc topples — unstable; if it is exactly on the axle, the disc stays in any position — neutral equilibrium. The reason lies in the energy: $U = M g y_{c}$ . In the stable position $U$ is at its lowest (the dip on the graph), in the unstable one at its highest (the peak), and in the neutral one it does not change. Without friction the body oscillates about the dip — mechanical energy is conserved; friction stops it at the minimum. The same is true of a body on a support: when the box on the board is tilted, its centre of gravity first rises (stable), and at $\tan θ = b / h$ the vertical line leaves the edge of the base — the equilibrium becomes unstable. Ways to increase stability: lower the centre of gravity (a ship's ballast, a crane's counterweight), widen the supporting area, and when suspending a body, place the axis above the centre of gravity.

Model simplifications

The bodies are rigid and uniform, and the problem is solved in a plane (2D). The loads on the disc hang vertically on their strings (they do not swing), and the friction at the axle is proportional to speed. Falling: rotation about the edge of the support ( $I \ddot{φ} = M g Δ x$ ), then free flight after ≈ 40°; once it has landed the body simply lies still (the impact is not modelled). The levels of the tower are identical cylinders. Sliding friction is $μ_{k} = 0.8 μ$ .

Did you know?

During the restoration work of 1990–2001 the lean of the Tower of Pisa was reduced from about 5.5° to 4°: soil was carefully removed from beneath the raised side of the tower.
The harmonic series grows without limit: with enough books, the top one can be pushed out beyond the edge of the table as far as you like. For example, to reach two book lengths you need 31 books.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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