Every real gear mesh and pulley wheel turns a little of the work into heat and sound. Students find a gear train lifts less than the ideal, that adding pulleys stops helping, work out overall efficiency by multiplying, and redesign a brief with friction in it.
The ideal block and tackle said more pulleys always means less pull. Real riggers stop at about six. Why?
The earlier Mechanisms boards had no friction, so work in = work out: a smaller force over a longer distance did exactly the same work. Real machines rub. Each gear mesh and each pulley wheel turns a little of the work into heat and sound, and the energy is still all there (it is conserved), but less of it is useful.
Efficiency = useful work out ÷ work in. Here each gear mesh passes on 95% of the turning force, and each pulley wheel passes on 90% of the pull.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
More and more pulleys
TryTry it on the board and compare the result with your prediction.
Try: friction in the winch and the pulleys
Was your prediction right? Past about 9 ropes, more pulleys stop helping.
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The winch's compound train (ratio 15) should lift 3000 g with no friction. Add 50 g at a time and lift, until it stalls.
Use the board. This ticks when you get there.
What is the heaviest bucket it lifted, in grams?
Pulleys with friction: find the fewest ropes that let your 150 N lift the 400 N sack.
Use the board. This ticks when you get there.
Now try 9 ropes, then 10. Compare the pulls.
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Worked example: multiply the losses
Worked exampleBefore each step, say what you think comes next.
0.9×0.9×0.9×0.9=0.66Each of the 4 wheels passes on 90% of what it gets: multiply.
pull=4×0.66400=153 NThe ideal pull (400 ÷ 4) divided by the efficiency.
work in=153×4=610 JYou still pull 4 m of rope to lift the sack 1 m.
work out=400×1=400 JThe useful work: the sack's weight times the height.
610−400=210 J of heat and soundEfficiency = 400 ÷ 610 = 66%.
Three gear meshes, each 95% efficient. What is the overall efficiency, to the nearest percent?
A machine is 80% efficient and must do 400 J of useful work. How much work must go in, in joules?
ModifyChange the problem on purpose and see what happens.
Vary: a brief with friction
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Need a hint?
Without friction, 900 ÷ 300 = 3 ropes would do. Try it.
Another hint
Each extra rope helps less. Keep adding and test each time.
Lift a 900 N engine with a pull of at most 300 N, using the fewest ropes. Each wheel is 90% efficient.
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Efficiency
Two stages, 90% and 80% efficient. What is the overall efficiency, in percent?
A winch is 75% efficient. It lifts a 600 N load 2 m. How much work goes in, in joules?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
The winch again, with friction
Need a hint?
With friction the train lifts 200 × ratio × 0.9 grams (two meshes).
Another hint
1500 ÷ (200 × 0.9025) ≈ 8.3: you need a ratio of at least 8.3.
Another hint
The time is 5 × ratio seconds, so 45 s allows a ratio up to 9. Which two stages multiply to between 8.3 and 9?
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Lift 1500 g within 45 s with a compound train, now that each mesh is 95% efficient.
Use the board. This ticks when you get there.
For teachers
Students meet friction in the gear train and pulley boards, find the winch lifts 2700 g instead of 3000 g and that pulleys beyond about nine stop helping, multiply stage efficiencies, account for the lost work as heat and sound, and redo two earlier briefs with friction in them. About 45 minutes.
I can work out a machine's overall efficiency by multiplying the efficiency of each stage.
I can explain where the 'lost' work goes, and why energy is still conserved.
I can explain why adding more pulleys or gears eventually stops helping.
I can redesign a machine so it still meets its brief with friction.
Before class
Do Gear trains and Pulleys first; Science › Year 9 Efficiency and Conservation are the Science homes of the ideas (the page links them).
The model: each gear mesh passes on a fixed fraction of the turning force (95%) and each pulley wheel a fixed fraction of the pull (90%), whatever the load; speed and rope length are unchanged. Real losses depend on lubrication, load and speed; 90–98% per stage is typical.
A hands-on check: a spring balance on a real block and tackle reads more than load ÷ ropes, and the gap grows with more pulleys.
Curriculum hooks: Design and Technologies, engineering principles and systems; Science, energy conservation and efficiency (Year 9); Maths, percentages and repeated multiplication.
Running the lesson
Predict: most expect 'keep getting smaller'. Ask why riggers stop at about six.
Try: have students say the frictionless answer before each test.
Worked example: build the 0.9 × 0.9 × … chain on the board and stop to ask what each 0.9 is 90% of.
Make: compare the frictionless and friction answers to the same brief.
The hard part (on purpose)
Students subtract the losses (95% − 5% − 5%) or think energy is destroyed. Keep asking 'of what?': each stage passes on 95% of what it received.
When students get stuck
A student subtracts the losses. Ask: The second mesh gets 95% of what? Fix: Of what the first passed on: multiply.
A student says friction destroys energy. Ask: What do the gears feel like after a long run? Fix: Warm: the energy became heat.
A student keeps adding pulleys. Ask: Did the last one make the pull smaller? Fix: Each extra wheel wastes more; past about 9 it costs more than it saves.
Stretch and support
Stretch: For 90% wheels, which number of ropes gives the smallest pull? Work out n × 0.9ⁿ for n = 8, 9, 10, 11.
Support: Give a table: ropes, ideal pull, efficiency (0.9ⁿ), real pull. Fill rows 1 to 4 together.
Where it leads
Next: Linkages and cams; Structures.
Science: Year 9 Efficiency and Where does the energy go? (the Builds on links on this page).
more and more pulleys: The frictionless rule (more ropes, less pull) invites 'add pulleys for ever'. Committing first makes the levelling-off a surprise worth explaining.
Try: friction in the winch and the pulleys: The gear train lifts less than its ideal 3000 g; the pulley's fewest ropes rise from 3 to 5, and ropes 9 and 10 give the same pull.
Worked example: multiply the losses: Each stage passes on a fraction of what it gets, so efficiencies multiply. Work in = work out ÷ efficiency, and the difference is the energy lost as heat and sound.
Vary: a brief with friction: Students redo a pulley brief with losses: the frictionless answer (3 ropes) now fails, and the rule of thumb becomes 'test it'.
efficiency: Multiply stage efficiencies, divide for work in, and the losses link to Science's efficiency.
the winch again, with friction: The Gear trains brief, redone with 95% per mesh: the frictionless answer (ratio 7.5) now stalls, so students need a slightly bigger ratio and a looser time limit.
Now you can
Work out a machine's overall efficiency by multiplying the efficiency of each stage.
Explain where the 'lost' work goes, and why energy is still conserved.
Explain why adding more pulleys or gears eventually stops helping.
Redesign a machine so it still meets its brief with friction.