Wheel and Axle

The wheel and axle is a lever bent into a circle. The arm lengths become radii, and the advantage is their ratio, which makes the arithmetic identical to the levers lesson.

The arrangement

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A wheel and a smaller axle are rigidly joined and turn together.

Mechanical advantage = wheel radius / axle radius.

In the figure, 12 inches over 3 inches gives an advantage of 4. A 400-pound load on the axle needs 100 pounds of effort at the wheel's rim.

It is a lever with the two arms being the two radii, and the axle's center is the fulcrum. That is why the arithmetic is the same and why the check is the same.

Diameters work equally well, since the ratio is unchanged. A question giving diameters does not need them halved first.

Which way round

Effort on the wheel, load on the axle: force is multiplied. Advantage above 1. A winch, a doorknob, a steering wheel.

Effort on the axle, load on the wheel: speed and distance are multiplied. Advantage below 1. A vehicle's driven wheels work this way - the axle turns and the wheel rim covers far more ground than the axle's surface does.

Both are real designs, and which one a question describes is decided by where the effort is applied.

The distance trade

The wheel's rim travels further than the axle's surface, in the same proportion as the radii.

With an advantage of 4, one turn moves the rope on the wheel four times as far as the rope on the axle. A quarter of the force over four times the distance, which is the same trade every machine makes.

The capstan

A capstan is a wheel and axle turned by bars pushed around it, winding rope on the drum. Its mechanical advantage is the bar length divided by the drum's radius: a 4-foot bar on a drum of 1-foot radius gives 4. Watch for a drum given as a diameter: a 6-inch diameter drum has a 3-inch radius, so a 36-inch bar gives 36 / 3 = 12.

Where it turns up

Anything with a crank or a handle is a wheel and axle, and this is the part that is not obvious.

A screwdriver is a wheel and axle, which is worth stating because it is asked and because it explains something practical: a fat-handled screwdriver turns a screw more easily than a thin one, because the wheel radius is larger for the same shank. Nothing about the blade changed.

A doorknob works the same way. Removing the knob and trying to turn the bare spindle is very hard, and the reason is the lost radius.

What you can skip

Across the 26 questions on this topic in our bank:

• Windlass and winch variants by name. The capstan is the one named, four times; the windlass never.
• Efficiency calculations come up once. The questions are ideal mechanical advantage: radius over radius.

Where people lose points

Inverting the ratio. Wheel over axle for a force advantage.

Halving a diameter unnecessarily. The ratio is the same either way.

Not recognizing a crank, a knob or a screwdriver as this machine.

Thinking the wheel and axle turn at different rates. They are joined, so they turn together; it is the surface speeds that differ.

Expecting an advantage above 1 when the effort is on the axle.

Work one in under a minute

A winch has a drum 4 inches in diameter and a crank handle that sweeps a circle 24 inches in diameter. What effort raises a 300-pound load?

Advantage: 24 over 4 is 6. Diameters are fine; no halving needed.

Effort: 300 divided by 6 is 50 pounds.

Check the distance: the handle travels six times as far as the rope winds in, so raising the load one foot means turning the handle through six feet of arc. The work is unchanged, as always.

Where this leads

This closes the six simple machines. Torque, the next lesson, is the quantity all the rotating ones are really about.