---
title: "Simple Machines: Levers, Pulleys, Inclined Planes"
book: "Primary & Middle School Physics"
subject: physics
language: en
chapter: 31
exercises: 11
source: https://one-course.com/books/physics/1/en/chapter/31-simple-machines-levers-pulleys-inclined-planes
---

# Chapter 31 — Simple Machines: Levers, Pulleys, Inclined Planes

No engine, no electricity — yet with a few beams, ropes and ramps, ancient builders raised stones that whole teams of oxen could not drag. Their toolbox held the *[simple machines](#def-g5-simple-machines-machine)*: inventions with no motor at all, which turn a weak, comfortable pull into a mighty one. You already own the first; meet the family.

## 31.1 The family of effort-savers

**Definition 31.1 (Simple machine).**

A *simple machine* is a device without a motor that changes a [force](https://one-course.com/books/physics/1/en/chapter/12-pushes-and-pulls-forces#def-g2-pushes-and-pulls-force) to suit us: making it stronger, or changing its direction, or both. The classics: the *[lever](https://one-course.com/books/physics/1/en/chapter/17-levers-and-balance-scales#def-g3-levers-and-scales-lever)* you know from the seesaw and crowbar, the *[pulley](#def-g5-simple-machines-pulley)*, and the *[inclined plane](#def-g5-simple-machines-ramp)* — the humble ramp.

**Example 31.2 (The lever, revisited).**

The [lever](https://one-course.com/books/physics/1/en/chapter/17-levers-and-balance-scales#def-g3-levers-and-scales-lever)’s rule you already own: coins times marks — your small [force](https://one-course.com/books/physics/1/en/chapter/12-pushes-and-pulls-forces#def-g2-pushes-and-pulls-force) far from the [pivot](https://one-course.com/books/physics/1/en/chapter/17-levers-and-balance-scales#def-g3-levers-and-scales-lever) beats a big load close to it. Notice something else now: your end of the crowbar sweeps a long way down while the boulder rises only a little. Keep that observation in your pocket; it is about to become a law.

## 31.2 Pulleys

**Definition 31.3 (Pulley).**

A *pulley* is a wheel with a groove for a rope. A *fixed* pulley hangs from a beam and only changes the pull’s direction: haul down, the load goes up. A *movable* pulley rides on the rope with the load hooked to it: the load then hangs from *two* strands of rope, and each strand carries only half the [weight](https://one-course.com/books/physics/1/en/chapter/24-weight-and-mass#def-g4-weight-and-mass-weight).

![Two pulleys, two gifts. The fixed pulley turns your pull around; the movable pulley lets two rope strands share the load, so your hand holds only half.](https://one-course.com/images/onecourse/chapters/physics-1/g5-simple-machines/fig-fabda52b6d28.svg)

*Two [pulleys](#def-g5-simple-machines-pulley), two gifts. The fixed [pulley](#def-g5-simple-machines-pulley) turns your pull around; the movable [pulley](#def-g5-simple-machines-pulley) lets two rope strands share the load, so your hand holds only half.*

![The well’s pulley: the rope rides in the groove, and a downward pull becomes an upward lift.](https://one-course.com/images/onecourse/chapters/physics-1/g5-simple-machines/img-3cbc3e8be90f.jpg)

*The well’s [pulley](#def-g5-simple-machines-pulley): the rope rides in the groove, and a downward pull becomes an upward lift.*

**Example 31.4 (Pulleys at work).**

The flag rises up its pole while your hands pull comfortably *down*: a fixed [pulley](#def-g5-simple-machines-pulley) at the top — direction changed, effort unchanged, but hauling down lets your own [weight](https://one-course.com/books/physics/1/en/chapter/24-weight-and-mass#def-g4-weight-and-mass-weight) help. The builder’s block-and-tackle chains several [pulleys](#def-g5-simple-machines-pulley) so the load hangs from four or six strands: each strand carries a quarter or a sixth of the [weight](https://one-course.com/books/physics/1/en/chapter/24-weight-and-mass#def-g4-weight-and-mass-weight), and one worker raises a piano. Watch the worker, though: [metre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) after [metre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) of rope races through their hands, while the piano creeps upward slowly.

## 31.3 The inclined plane

**Definition 31.5 (Inclined plane).**

An *inclined plane* — a ramp — is a slope for raising loads without lifting them straight up. The gentler the slope, the smaller the push needed to move the load along it: rolling a barrel up a long, gentle ramp is easy; up a short, steep one, brutal; straight up, impossible.

**Method 31.6 (Feel it with the rubber-band scale).**

Your rubber-band scale from last year can taste the difference:

1. load a toy car onto a plank and hook the rubber band to it;
2. prop the plank as a gentle ramp onto a low book pile; drag the car slowly up by the band and note the stretch;
3. prop the same plank much steeper, onto a chair seat, and drag again.

The steep drag stretches the band far more: steeper slope, bigger effort. Gentle slopes really are gentle — your instrument says so.

![Same crate, same height to gain: the long gentle ramp asks a small push over a long road; the steep ramp, a big push over a short one.](https://one-course.com/images/onecourse/chapters/physics-1/g5-simple-machines/fig-459434e0ba3b.svg)

*Same crate, same height to gain: the long gentle ramp asks a small push over a long road; the steep ramp, a big push over a short one.*

**Example 31.7 (Ramps in the landscape).**

Wheelchair ramps run long and shallow beside short staircases — gentleness on purpose. Mountain roads zigzag: rather than attack the summit straight up, they wrap a long, gentle [inclined plane](#def-g5-simple-machines-ramp) back and forth across the slope. And the screw is a ramp in disguise — a slope wrapped around a rod, so each easy turn of the screwdriver sneaks the screw a tiny step deeper.

## 31.4 The golden rule

**Proposition 31.8 (The golden rule of machines).**

No [simple machine](#def-g5-simple-machines-machine) gives something for nothing: whatever it saves in [force](https://one-course.com/books/physics/1/en/chapter/12-pushes-and-pulls-forces#def-g2-pushes-and-pulls-force), it charges back in distance. Half the [force](https://one-course.com/books/physics/1/en/chapter/12-pushes-and-pulls-forces#def-g2-pushes-and-pulls-force) — twice the rope hauled, or twice the road traveled; a tenth of the [force](https://one-course.com/books/physics/1/en/chapter/12-pushes-and-pulls-forces#def-g2-pushes-and-pulls-force) — ten times the road. Machine after machine, the trade is exact: comfort is bought with [metres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units).

**Example 31.9 (Checking the rule).**

The movable [pulley](#def-g5-simple-machines-pulley): your hand pulls with half the [force](https://one-course.com/books/physics/1/en/chapter/12-pushes-and-pulls-forces#def-g2-pushes-and-pulls-force), and to raise the load $1\,\mathrm{m}$ you haul in $2\,\mathrm{m}$ of rope — both strands must shorten by a [metre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units). The crowbar: your end sweeps a long arc, the boulder barely stirs. The ramp: the crate reaches the same height, but you pushed it along the whole slope. Every saving paid for, precisely.

**Remark 31.10 (Why the rule cannot be cheated).**

Hiding under the golden rule is our old friend from the [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) chapter: lifting a load to a height costs a fixed helping of [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy), and no arrangement of ropes and planks changes the price. Machines only let you pay it in smaller coins — more of them. A small [force](https://one-course.com/books/physics/1/en/chapter/12-pushes-and-pulls-forces#def-g2-pushes-and-pulls-force) over a long road hands over the same [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) as a big [force](https://one-course.com/books/physics/1/en/chapter/12-pushes-and-pulls-forces#def-g2-pushes-and-pulls-force) over a short one. The forever-machines failed for this very reason: the bill is always the bill.

## 31.5 Exercises

**Exercise 31.1 ★.**

Name the three classic [simple machines](#def-g5-simple-machines-machine), and one everyday place each is found.

**Solution of Exercise 31.1.**

The [lever](https://one-course.com/books/physics/1/en/chapter/17-levers-and-balance-scales#def-g3-levers-and-scales-lever) (crowbar, seesaw, scissors), the [pulley](#def-g5-simple-machines-pulley) (flagpole, well, crane), the [inclined plane](#def-g5-simple-machines-ramp) (wheelchair ramp, mountain road, loading ramp).

**Exercise 31.2 ★.**

What does a fixed [pulley](#def-g5-simple-machines-pulley) change about your pull? What does a movable [pulley](#def-g5-simple-machines-pulley) change?

**Solution of Exercise 31.2.**

The fixed [pulley](#def-g5-simple-machines-pulley) changes only the direction of the pull (haul down, load goes up). The movable [pulley](#def-g5-simple-machines-pulley) changes its size: the load hangs from two strands, so the hand holds only half the [weight](https://one-course.com/books/physics/1/en/chapter/24-weight-and-mass#def-g4-weight-and-mass-weight).

**Exercise 31.3 ★.**

Why is hauling a flag *down*ward to raise it more comfortable than lifting the same [weight](https://one-course.com/books/physics/1/en/chapter/24-weight-and-mass#def-g4-weight-and-mass-weight) straight up, even though the effort is the same size?

**Solution of Exercise 31.3.**

Pulling down recruits your own [weight](https://one-course.com/books/physics/1/en/chapter/24-weight-and-mass#def-g4-weight-and-mass-weight): you can lean or even hang on the rope, letting the Earth’s pull help your muscles. Lifting upward, your arms fight alone.

**Exercise 31.4 ★.**

With one movable [pulley](#def-g5-simple-machines-pulley), the load hangs from two strands. What share of the [weight](https://one-course.com/books/physics/1/en/chapter/24-weight-and-mass#def-g4-weight-and-mass-weight) does your hand hold? To raise the load $3\,\mathrm{m}$, how much rope must you haul?

**Solution of Exercise 31.4.**

Half the [weight](https://one-course.com/books/physics/1/en/chapter/24-weight-and-mass#def-g4-weight-and-mass-weight). To raise the load $3\,\mathrm{m}$, both strands must shorten by $3\,\mathrm{m}$: your hand hauls $2 \times 3 = 6\,\mathrm{m}$ of rope.

**Exercise 31.5 ★.**

In [Method 31.6](#met-g5-simple-machines-measure), which ramp stretches the rubber band more? What pays for the gentler ramp’s smaller stretch?

**Solution of Exercise 31.5.**

The steep ramp stretches it more. The gentle ramp’s smaller effort is paid for in distance: the same height is gained along a much longer road — the golden rule’s trade.

**Exercise 31.6 ★.**

Why do mountain roads zigzag instead of running straight up the slope? Which [simple machine](#def-g5-simple-machines-machine) is the whole road?

**Solution of Exercise 31.6.**

Straight up the slope, the pull needed would defeat any engine or mule team; the zigzags wrap a long, gentle [inclined plane](#def-g5-simple-machines-ramp) across the mountainside — small effort, long road. The whole road is one stretched-out ramp.

**Exercise 31.7 ★.**

State the golden rule of machines. A machine lets you pull with a tenth of the [force](https://one-course.com/books/physics/1/en/chapter/12-pushes-and-pulls-forces#def-g2-pushes-and-pulls-force) — what does it charge you?

**Solution of Exercise 31.7.**

No machine gives something for nothing: [force](https://one-course.com/books/physics/1/en/chapter/12-pushes-and-pulls-forces#def-g2-pushes-and-pulls-force) saved is repaid in distance, exactly. A tenth of the [force](https://one-course.com/books/physics/1/en/chapter/12-pushes-and-pulls-forces#def-g2-pushes-and-pulls-force) costs ten times the road (or rope).

**Exercise 31.8 ★.**

How is a screw an [inclined plane](#def-g5-simple-machines-ramp) in disguise? What plays the role of the long gentle road?

**Solution of Exercise 31.8.**

Its thread is a long gentle slope wrapped around the rod. The many easy turns of the screwdriver are the long road; the tiny sinking of the screw each turn is the small height gained.

**Exercise 31.9 ★★.**

A block-and-tackle hangs a piano from four strands. What fraction of the piano’s [weight](https://one-course.com/books/physics/1/en/chapter/24-weight-and-mass#def-g4-weight-and-mass-weight) does the worker’s hand pull? To raise the piano $2\,\mathrm{m}$, how many [metres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) of rope pass through their hands?

**Solution of Exercise 31.9.**

A quarter of the [weight](https://one-course.com/books/physics/1/en/chapter/24-weight-and-mass#def-g4-weight-and-mass-weight). Each of the four strands must shorten by $2\,\mathrm{m}$: $4 \times 2 = 8\,\mathrm{m}$ of rope pass through the worker’s hands.

**Exercise 31.10 ★★.**

A loading ramp is $6\,\mathrm{m}$ long and rises $1\,\mathrm{m}$; pushing a crate up it takes a modest push. The impatient movers cut a new ramp only $3\,\mathrm{m}$ long to the same platform. Using the golden rule, compare the new push with the old one — and explain why “half the road” was no bargain.

**Solution of Exercise 31.10.**

The new ramp climbs the same $1\,\mathrm{m}$ in half the road, so it is twice as steep, and the push roughly doubles. The golden rule collected instantly: road halved, [force](https://one-course.com/books/physics/1/en/chapter/12-pushes-and-pulls-forces#def-g2-pushes-and-pulls-force) doubled — the crate’s climb costs the same either way, and the movers merely traded many comfortable coins for a few heavy ones.

**Exercise 31.11 ★★.**

An advertisement offers a [pulley](#def-g5-simple-machines-pulley) set “so clever that you pull one [metre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) of rope with half the [force](https://one-course.com/books/physics/1/en/chapter/12-pushes-and-pulls-forces#def-g2-pushes-and-pulls-force) — and the load rises a full [metre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units)”. Put the claim on trial before the golden rule and the [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) bill of [Remark 31.10](#rem-g5-simple-machines-whygolden): could any arrangement of wheels and ropes honor it?

**Solution of Exercise 31.11.**

No arrangement can honor it. Raising the load one [metre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) costs a fixed helping of [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy); paying with half the [force](https://one-course.com/books/physics/1/en/chapter/12-pushes-and-pulls-forces#def-g2-pushes-and-pulls-force), the seller must haul *two* [metres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) of rope — [half-force](https://one-course.com/books/physics/1/en/chapter/12-pushes-and-pulls-forces#def-g2-pushes-and-pulls-force) along just one [metre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) hands over only half the bill. The advertisement promises a paid lift at half price: exactly the free lunch that the golden rule, and the failed forever-machines, forbid.