Useful work and total work
When you use a machine, you do work on it. This is the total work (work input) = effort × distance the effort moves.
The machine does work on the load. This is the useful work (work output) = load × distance the load moves.
- An ideal machine has no friction and no heavy parts. Useful work = total work.
- In a real machine some work is wasted: against friction (as heat and sound) and in lifting the moving parts of the machine. So useful work is less than total work.
Energy is never lost, it just goes to places where we do not want it. Wasted work = total work − useful work.
η: calculating efficiency
Efficiency (η, say "eta") = useful work ÷ total work × 100%.
It tells us what part of the work you put in becomes useful. It has no unit.
Steps
- Find useful work: load × height lifted.
- Find total work: effort × distance the effort moves.
- Divide useful by total, multiply by 100.
Example: useful work 60 J, total work 80 J. η = 60 ÷ 80 × 100 = 75%.
η is always less than 100% for a real machine. A value above 100% means a mistake.
Link to mechanical advantage
Efficiency = actual MA ÷ ideal MA × 100. A pulley block with 4 strands has ideal MA 4. If it really gives MA 3, then η = 3 ÷ 4 = 75%.
Improving efficiency
To waste less, cut the two big losses.
- Reduce friction: oil or grease the axles and ropes; use smooth surfaces, ball bearings and wheels.
- Use lighter moving parts: a light pulley block, a light bucket, a light chain.
- Keep machines clean and well fitted so that parts do not rub.
- Lift bigger loads at one time (up to the safe limit): the fixed waste is then small compared with the useful work.
Remember: no machine has 100% efficiency. Even the best gear box loses some energy as heat.
Try it: raise the green, shrink the red
In the last 3D step, set load 60 N, pulley 10 N and friction 5 N (η = 75%). Now set friction to 0. What is η? (60 ÷ 70 = 86%.) Now set the pulley weight to 0 too. (100%.) Then double the load with pulley 10 N and friction 5 N: does η go up or down?
At home
Slide a book down a rough ramp and a smooth ramp. Which one wastes more energy as heat? Rub your palms for 10 seconds: that warm feeling is wasted work.
Key formulas and definitions
- Useful work (output) = load × distance moved by the load
- Total work (input) = effort × distance moved by the effort
- Wasted work = total work − useful work
- Efficiency η = (useful work ÷ total work) × 100%
- η = (actual MA ÷ ideal MA) × 100%
- For a movable pulley (2 strands): effort = (load + pulley weight) ÷ 2 + friction
Worked examples
1. A machine does 80 J of total work and 60 J of it is useful. Find η and the wasted work.
η = 60 ÷ 80 × 100 = 75%. Wasted = 80 − 60 = 20 J.
2. A pulley system lifts a 50 N load by 2 m. The effort is 30 N and it moves 4 m. Find the efficiency.
Useful = 50 × 2 = 100 J. Total = 30 × 4 = 120 J. η = 100 ÷ 120 × 100 = 83.3%.
3. A crate of 500 N is raised 1 m by pushing it up a 5 m ramp with 150 N. Find the efficiency.
Useful = 500 × 1 = 500 J. Total = 150 × 5 = 750 J. η = 500 ÷ 750 × 100 = 66.7%. The rest is lost to friction.
4. A pump has 60% efficiency. It takes 800 J of energy. How much is useful and how much is wasted?
Useful = 60% of 800 = 480 J. Wasted = 800 − 480 = 320 J.
5. A pulley block has 4 strands. It lifts a 200 N load with a 70 N effort. Find the efficiency.
For 1 m lift: useful = 200 J. Rope pulled = 4 m, total = 70 × 4 = 280 J. η = 200 ÷ 280 × 100 = 71.4%. Or: actual MA = 200 ÷ 70 = 2.86, ideal MA = 4, η = 2.86 ÷ 4 = 71.4%.
6. A movable pulley (2 strands) weighs 10 N and has 5 N of friction. It lifts a 90 N load. Find the effort and the efficiency. Then find the efficiency for a 190 N load.
Effort = (90 + 10) ÷ 2 + 5 = 55 N. For a 1 m lift: useful = 90 J, total = 55 × 2 = 110 J. η = 90 ÷ 110 = 81.8%. For 190 N: effort = (190 + 10) ÷ 2 + 5 = 105 N, total = 210 J, η = 190 ÷ 210 = 90.5%. A bigger load makes the machine more efficient.
Common mistakes
- Using the load as the useful work. Useful work is load × distance, not just load.
- Putting total work in the top and useful in the bottom (η above 100%). Useful is always on top.
- Forgetting to multiply the effort by the distance the EFFORT moves (rope pulled), not the load distance.
- Thinking high mechanical advantage means high efficiency. They are different things.