Work and Energy - CourseWiki

Work is a force moving something through a distance

Holding a heavy box still is exhausting, and in this sense it is zero work. Work needs a component of force along the displacement; work is negative when that component opposes the movement.

W = Fs — work equals force times displacement for a constant force parallel to the motion.

Work is one of the ways energy moves from one place to another, so it is measured in energy units and is often written E when what matters is the energy delivered rather than the act of delivering it.

The two systems, for work and every other form of energy.
System	Unit	Symbol	Built from
SI	joule	`J`	`1 J = 1 N·m`
Foot–slug–second	foot pound-force	`ft·lbf`	`1 ft·lbf = 1.355818 J`

Worked example: lifting a load

How much work is required to lift a 2,000 lb load to a height of 10 ft?

The force needed to lift steadily is the load’s weight, 2,000 lbf, and it acts straight up along the 10 ft of travel.
W = 2,000 lbf × 10 ft = 20,000 ft·lbf.
In SI: 20,000 ft·lbf × 1.355818 J per ft·lbf = 27,116 J, or 27.1 kJ.

Two things this calculation assumes, both worth saying out loud: the load is raised at a steady speed, so no extra force goes into accelerating it, and the 2,000 lb figure is a weight in pound-force, not a mass in pound-mass. On Geometry, Mass, and Force those are separate quantities, and here it is the force that belongs in the formula.

Work and torque share dimensions - word order is a convention; Watch out

Work and torque share dimensions - word order is a convention

Multiply a constant force by displacement along that force and you get work. Multiply a force by a radius and you get torque. Both are a force times a length, so in the foot–pound system both come out as some arrangement of feet and pounds-force.

Here torque is written lbf·ft. Energy is written ft·lbf. The numerical factor is identical — 1.355818 of either — but word order alone does not establish the physical quantity. Torque describes turning effort; work describes energy transferred through displacement.

SI removes the ambiguity by naming the energy unit separately. A torque stays N·m; energy becomes the joule. Numerically 1 J = 1 N·m, but a torque is never reported in joules. See Torque.

Turning a torque into work, and where the radian goes

A constant torque about a fixed axis that turns through an angle does work: W = τθ, with the angle in radians. Because the radian is a ratio of two lengths, NIST gives its base-unit expression as m/m and its value as 1. Nothing is added to the units, so:

1 J = 1 N·m (torque) × 1 rad · 1 ft·lbf = 1 lbf·ft (torque) × 1 rad

This is the practical payoff of the radian being dimensionless: a motor rated in N·m turning through an angle in rad gives you joules directly, with no factor to look up. Use degrees or revolutions instead and the relationship breaks, because both of those carry a chosen count inside them.

The energy units you will actually meet

Energy is billed, rated, and specified in a lot of different units. They all convert; the only work is knowing which one a number is in.

Common energy units, expressed in joules.
Unit	Symbol	In joules	Where it turns up
joule	`J`	1 J	The SI unit. Everything else converts to it.
foot pound-force	`ft·lbf`	1.355818 J	U.S. mechanical work and energy.
watt hour	`W·h`	3,600 J (exact)	Battery capacity.
kilowatt hour	`kW·h`	3,600,000 J (exact)	The line on an electricity bill.
British thermal unit (IT)	`Btu_IT`	1,055.056 J	Heating, cooling, and fuel ratings.
horsepower hour	`hp·h`	≈ 2,684,520 J	Engine and drive energy.
calorie (IT)	`cal_IT`	4.1868 J (exact)	Thermal work and material data.

A kilowatt hour is a power multiplied by a time, and that is all it is. One kilowatt sustained for one hour. It is not an SI unit — NIST lists it among the units to be converted into SI — but it is exactly 3.6 MJ, and it is the unit an electricity meter counts in.

“Btu” on its own is ambiguous. NIST lists six different British thermal units, from 1,054.35 J to 1,059.67 J depending on which definition is meant. The spread is about half a percent, which is small until it is multiplied by a fuel bill. Use Btu_IT unless a specification names another one.

Sources

Where these values were checked

All conversions from NIST SP 811, Appendix B: foot pound-force to joule 1.355 818 E+00; watt hour to joule 3.6 E+03 (exact); kilowatt hour to joule 3.6 E+06 (exact); British thermal unit_IT to joule 1.055 056 E+03, with Btu_th 1.054 350 E+03, Btu (mean) 1.055 87 E+03, Btu (39 °F) 1.059 67 E+03, Btu (59 °F) 1.054 80 E+03 and Btu (60 °F) 1.054 68 E+03 listed separately; calorie_IT to joule 4.1868 E+00 (exact).

The horsepower hour figure is derived rather than quoted: one horsepower is 7.456 999 E+02 watts in the same table, and an hour is 3,600 s, so 745.6999 W × 3,600 s = 2,684,520 J to six figures.

NIST SP 811, Chapter 4, Table 3, gives the joule as J, expressed as N·m and in base units as m²·kg·s⁻², and the radian as rad, value 1, base-unit expression m/m.