Mechanical Advantage

This is the organizing idea for the whole subtest. Every simple machine in the lessons that follow is one way of making the same trade, so understanding it once saves learning six separate things.

The trade

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Look at the numbers in the figure. A 100-pound effort lifts a 400-pound load, which sounds like something for nothing. It is not: the effort moves four inches while the load rises one.

• Work in: 100 pounds through 4 inches is 400 inch-pounds.
• Work out: 400 pounds through 1 inch is 400 inch-pounds.

Equal. The machine multiplied the force by four and divided the distance by four, and the work is unchanged.

That is what every machine does. It cannot do otherwise, because energy is conserved.

Calculating it

Mechanical advantage = load / effort, or equivalently effort distance / load distance.

It has no units, because it is a ratio.

A pulley system lifts 600 pounds with 150 pounds of effort.

600 divided by 150 is 4.

A machine's input moves 30 inches while its output moves 6.

30 divided by 6 is 5.

Both forms give the same answer for an ideal machine, which is the check worth running when a question supplies both.

Reading the number

An advantage below 1 is not a broken machine. Sometimes speed or distance is what you want, and force is what you are prepared to spend. A bicycle in a high gear, and the human forearm, both work that way - the muscle pulls hard over a short distance so the hand moves fast over a long one.

A fixed pulley has an advantage of 1 and is still useful, because changing the direction of a pull is worth having.

Ideal against actual

Ideal mechanical advantage (IMA) comes from the geometry: the distances, the arm lengths, the number of rope segments.

Actual mechanical advantage (AMA) is what you measure, and it is always less, because friction takes a share.

Efficiency = AMA / IMA, as a percentage, or equivalently work out over work in.

No machine is 100 percent efficient. Friction converts part of the input to heat, always. A question offering an efficiency above 100 percent is offering something impossible, and a question asking why actual falls short of ideal wants friction.

The six simple machines

Every one makes the same trade, and each has its own lesson.

Wedges and screws are inclined planes in disguise, which reduces six machines to four ideas.

Compound machines

Machines can be combined, and the advantages multiply.

A system with a lever of advantage 3 feeding a pulley of advantage 4 gives a total of 12.

The trade multiplies too, so the effort in that system moves twelve times as far as the load does. Nothing is escaped by combining machines; the trade just gets steeper.

What you can skip

Across the 27 questions on this topic in our bank:

• Velocity ratio never appears by name. It is the same distance ratio this lesson calls ideal mechanical advantage.
• Actual mechanical advantage is named once; efficiency twice.

Where people lose points

Thinking a machine reduces the work required. It reduces the force and increases the distance.

Believing a machine can create energy.

Treating an advantage below 1 as useless. It buys speed or distance.

Saying a fixed pulley is pointless because its advantage is 1. It changes direction.

Forgetting friction when asked why actual falls short of ideal.

Dividing effort by load. It is load over effort.

Work one in under a minute

A ramp lets a worker push a 500-pound crate up with 125 pounds of force. What is the mechanical advantage, and if the ramp is 12 feet long, how high is it?

Advantage: 500 divided by 125 is 4.

For an ideal machine the distance ratio matches the force ratio, so the ramp length is four times the height.

12 divided by 4 is 3 feet high.

Check the work: 125 pounds over 12 feet is 1,500 foot-pounds in; 500 pounds over 3 feet is 1,500 foot-pounds out. Equal, as it must be.

Where this leads

Every machine lesson that follows is this idea in a particular shape, so the arithmetic here is the arithmetic there.