Power Measurement

Part of Simple Machines

How to quantify mechanical power output and compare different power sources and machines.

Why This Matters

Power is the rate at which work is done. Force alone tells you how hard you push; power tells you how fast you can push that hard continuously. A person can exert enormous force briefly (jumping, throwing) but only moderate force continuously (walking, pulling). A horse can exert both moderate force and moderate continuous power — far more sustained power than a human, which is why draft animals transformed agriculture.

Measuring power allows you to make quantitative decisions: Is one water wheel enough to run two millstones? Can this horse team pull the stone block up the hill in a reasonable time? What is the effective power of the windmill, and how many workshops can it run? Without measurement, these are guesses. With measurement, they are engineering decisions.

The concept of horsepower — a unit of power that predates electrical measurement — was invented specifically to allow comparisons between steam engines and the horses they were replacing. Understanding the origin and meaning of power units, and how to measure them with simple tools, gives you genuine capability to quantify and compare your community’s mechanical resources.

Fundamental Concepts

Work: Force applied over a distance.

Work (kg⋅m) = Force (kg) × Distance (m)

Power: Rate of doing work.

Power (kg⋅m/s) = Work / Time = (Force × Distance) / Time = Force × Velocity

Horsepower: Defined by James Watt in the 1780s as 33,000 foot-pounds per minute, equivalent to 550 foot-pounds per second, or approximately 76 kg⋅m/s.

Watt: The SI unit of power: 1 Watt = 1 joule/second = approximately 0.102 kg⋅m/s.

Practical conversions:

• 1 horsepower (HP) = 76 kg⋅m/s = 746 Watts
• 1 human sustained effort ≈ 75-100 Watts = approximately 0.1 HP
• 1 horse sustained effort ≈ 500-750 Watts = 0.7-1.0 HP

Measuring Human Power Output

Step test method: A staircase or set of steps provides a simple power measurement for human subjects.

1. Measure the vertical height of the steps (h, in meters)
2. Have the person step up and down the steps at a steady pace for 3 minutes
3. Count the total number of steps up (n)
4. Measure the person’s body weight (m, in kg)
5. Calculate: Power = (m × g × h × n) / time Simplifying with g ≈ 1 (in kg⋅m/s units): Power ≈ (m × h × n) / t

Example: A 70 kg person climbs 60 steps of 0.2 m height each in 3 minutes:

Power = (70 × 0.2 × 60) / 180 = 840/180 = 4.7 kg⋅m/s = 46 Watts

This is modest (about 0.06 HP) — reflecting that this person is doing moderate aerobic work. At peak effort: 200-400 Watts for 30-60 seconds.

Rope-pull test:

1. Set up a pulley and known weight
2. Have the person pull the rope, lifting the weight steadily
3. Measure pull force (at the hauling end of the rope) and pull velocity (rope speed)
4. Power = Force × velocity

For a block and tackle with 4:1 MA lifting a 200 kg weight: the person pulls 50 kg at a rope speed of 1 m/s:

Power (at the hauling end) = 50 × 1.0 = 50 kg⋅m/s ≈ 490 Watts
But the load moves at 1/4 m/s, and output power = 200 × 0.25 = 50 kg⋅m/s (same, minus friction)

Measuring Animal Power Output

Standard test for draft animals (Prony brake equivalent):

A simple measurement of sustained pulling power:

1. Have the animal pull a known force on a flat road
2. Measure the force with a simple spring scale or balance in the trace
3. Time the animal walking a known distance
4. Calculate pull speed: distance/time
5. Power = force × speed

Example: A horse pulls 120 kg on level ground at walking pace (1.4 m/s):

Power = 120 × 1.4 = 168 kg⋅m/s = 1,650 Watts = 2.2 HP

This is a typical working horse at moderate effort.

For a full working day (8 hours): A horse can sustain approximately 0.7-1.0 HP continuously for a working day. Higher power is available for shorter periods (pulling a heavy load up a hill) but cannot be maintained.

Measuring Water Wheel Power

Field measurement:

A working water wheel’s power output can be estimated from:

1. Flow rate: Measure the volume of water passing the wheel per second
   • For a small stream: dam the channel briefly, divert flow into a container of known volume, time how long to fill
   • For a larger stream: measure channel width × depth × flow velocity
2. Head (height): The vertical drop of water that the wheel captures
   • For overshot wheels: measure the height from the point where water hits the wheel to the tailrace
   • For undershot wheels: measure the velocity of water and use kinetic energy formula
3. Efficiency factor: Typical water wheel efficiency 50-80%

Theoretical power (overshot) = Flow (kg/s) × Head (m) × Efficiency

Example: 50 liters/second flow (50 kg/s), 2.5 m head, 65% efficiency:

Power = 50 × 2.5 × 0.65 = 81 kg⋅m/s = 795 Watts ≈ 1.1 HP

Torque measurement method:

1. Attach a braking device to the wheel shaft: a rope wrapped around the shaft drum, with a known weight hanging from one end and a spring scale on the other end
2. Measure the shaft rotation speed (count revolutions per minute)
3. Net torque = (hung weight − spring scale reading) × drum radius
4. Power = Torque × angular velocity = Torque × (2π × rpm / 60)

This is the historical Prony brake method — the standard way of measuring engine power before electronic instruments.

Comparing Power Sources

Typical power outputs for planning purposes:

Power Source	Sustained Output	Notes
Human, light work	60-100 W	8-hour sustainable
Human, heavy work	150-200 W	4-hour sustainable
Human, peak effort	400-600 W	30-60 seconds only
Donkey, working	200-400 W	8 hours sustainable
Ox, working	400-600 W	8 hours sustainable
Horse, working	500-750 W	8 hours sustainable
Draft horse team (2)	900-1,400 W	Matched pair
Small waterwheel (2m, good site)	500-2,000 W	Continuous, no fuel
Medium waterwheel (4m)	2,000-8,000 W	Site dependent
Small windmill (6m span)	500-2,000 W	Variable, intermittent
Large windmill (12m span)	2,000-10,000 W	Variable, intermittent

Key insight: A well-sited water wheel of modest size outperforms a horse team at the same sustained work, continuously and without food cost. Investing in water power infrastructure has very high long-term returns.

Power per Person-Hour: Economic Comparison

For community planning, express power as the amount of work done per person-hour of labor investment:

Method	Power per Person (output)	Labor Required	Cost per Output Unit
Human direct labor	100 W/person	1 person/100 W	Very high
Animal with driver	600 W + care	0.5 person/hour	Moderate
Water wheel (once built)	3,000 W	0.1 person/hour maintenance	Very low

This comparison explains why civilizations invested in water mills: one person managing a water mill could do the work of 30-50 people grinding grain by hand. The mill freed those people for other productive activities.

Using Power Measurements for Machine Design

Matching power source to load:

Calculate the power demand of your intended machine, then verify your power source can supply it.

Example: Planning a grain mill:

1. Determine required millstone speed: 120 rpm
2. Determine required millstone torque: approximately 50 kg⋅m for a 60 cm stone
3. Required power = torque × angular velocity = 50 × (2π × 120/60) = 50 × 12.6 = 628 W
4. Your water wheel produces 800 W (from measurement)
5. After gear train friction (85%): available power = 800 × 0.85 = 680 W
6. 680 W available > 628 W required — the wheel is sufficient with small margin

If the calculation showed insufficient power, you would need to either increase the water head, widen the wheel, or reduce the millstone size.