Physics · Mechanics · Grade 10

Simple machines: lever, pulley

Why doesn't a crane tip over, and how does a pulley reduce the force?

The situation

On a building site in Tashkent a tower crane carries concrete blocks weighing several tonnes out to 40 metres. The crane itself is heavy, yet its base is only a few metres wide. Why doesn't it tip over? Down below, a worker easily hauls up a sack of cement with a simple system of pulleys — a block and tackle. How can a small force lift a heavy load, and what is the “price”?

Topic

Simple machines: lever and pulley. Construction crane: a small force lifts a heavy load

Key ideas

Moment of a force
Equilibrium of a lever
Golden rule of mechanics

Formulas

M = F d

Moment of a force: force × lever arm (the distance from the axis of rotation to the line of action of the force).

m_{q} g c \approx m g d + m_{s} g x_{s}

The crane is a lever: the moment of the counterweight balances the moments of the load and the jib. The tower is the pivot.

| M | \leq M_{h o l d} = m_{t o t} g a

Condition for not tipping over: the resultant moment must not exceed the moment that the whole weight of the crane can hold at a lever arm of $a = 2$ m.

F = \frac{m g}{n} (ideal), η = \frac{m g h}{F s}

Block and tackle: the load is shared among $n$ rope strands. With friction, $F$ is larger and the efficiency is $η < 1$ .

s = n h, F s = m g h

The golden rule of mechanics: however many times we gain in force, we lose the same number of times in distance — there is no gain in work.

Theory

The crane — a giant lever

The jib and counter-jib (the rear part) of a tower crane form a lever that can rotate about the tower. Each weight produces a turning moment: $M = F d$ . The load turns it to the right, the counterweight to the left. If the moments are equal, the lever is in equilibrium. The farther away the load, the greater its moment — which is why a crane can lift only a light load at the tip of the jib.

Why doesn't the crane tip over?

The crane's base is $2 a = 4$ m wide. While the centre of gravity of the whole crane stays inside the base, gravity holds the crane in place. If the resultant moment exceeds $m_{t o t} g a$ , the centre of gravity moves beyond the edge of the base and the crane tips over about that edge. In the scene, the sphere and the point dropped from it to the ground mark the crane's centre of gravity. Moving the counterweight too far out is also dangerous: a crane with no load tips over backwards.

Pulley and block and tackle

A fixed pulley gives no gain in force — it only changes the direction of the force. A movable pulley shares the load between two rope strands. In a block and tackle made of several pulleys, $n$ strands support the load and the tension in each is $m g / n$ . But to raise the load by $h$ , each strand must shorten by $h$ — so a length of rope $s = n h$ has to be pulled.

Moment as a vector; a couple

The moment of a force is not just a number but a vector: $\vec{M} = \vec{r} \times \vec{F}$ . Its magnitude is $F d$ ( $d$ is the lever arm of the force, the shortest distance from the axis to the line of action of the force), and its direction is along the axis of rotation, given by the right-hand screw rule: if a screw is turned in the direction the force turns things, it advances in the direction of $\vec{M}$ . In the crane, the load turns the jib clockwise and the counterweight turns it the other way — their moment vectors point in opposite directions and their sum $\sum \vec{M}$ must be small (switch on “Moment vectors”). A couple is a pair of parallel forces that are equal in magnitude, opposite in direction and do not lie on one line: turning a steering wheel with two hands, a spanner on a nut, the crane handle. Their resultant is zero, so they do not move the body, they only rotate it: the moment $M = F d$ ( $d$ is the distance between the lines of action) is the same about any point.

Other mechanisms and the dynamics of tipping

Simple machines based on the principle of moments: the lever (scissors, crowbar, hand cart), the windlass (a well winch) — a small force at a large radius balances a large force at a small radius — and gear and chain drives (a bicycle). A pulley is also a lever: a fixed pulley has equal arms, a movable pulley has arms in the ratio 1 : 2. “Build” a simple machine yourself: choose the number of strands in the block and tackle, and measure the force and the distance. If the balance is lost, the crane tips about the edge of its base according to the law of rotational dynamics: $I ε = M$ , where $I = \sum m_{i} r_{i}^{2}$ is the moment of inertia about the edge. For example, if we treat the tower as a rod, then about an axis through its lower end, by the parallel-axis theorem, $I = \frac{1}{12} m H^{2} + m {(\frac{H}{2})}^{2} = \frac{1}{3} m H^{2}$ (details — in the “Dynamics of a rigid body” experiment).

The golden rule and the model

In an ideal machine the work done $F s$ equals the useful work $m g h$ : simple machines give no gain in work. With friction, each pulley transmits about 95 % of the tension, the work expended is greater and the efficiency is below 100 %. Simplifications of the model: the masses of the crane parts are approximate, the crane is a rigid body, there is no wind; tipping is rotation about the edge of the base; the masses of the pulleys and the rope are neglected.

Did you know?

According to legend, Archimedes said about the lever: “Give me a place to stand, and I will move the Earth.” These words were quoted by later Greek scholars (for example, Pappus of Alexandria).
The Roman architect Vitruvius (1st century BC), in his “Ten Books on Architecture”, described hoisting machines with three pulleys (trispastos) and five pulleys (pentaspastos). The word “polyspast” comes from the Greek for “pulled many times”.

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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