Mechanical Advantage Calculator

What the mechanical advantage calculator does

A simple machine lets you move a big load with a small effort, and mechanical advantage is the number that says by how much. This calculator works it out for the six classic simple machines: the lever, the pulley, the screw, the wedge, the inclined ramp, and the wheel and axle. Pick the machine, enter its dimensions, and it returns the factor by which your force is multiplied.

Below is the trade that every machine makes, the formula for each one, and a worked example.

How to use it

1. Choose a machine from the menu.
2. Enter its dimensions. A lever needs its two arm lengths, a ramp its height and length or its angle, a wheel and axle their two radii, and so on.
3. Press Calculate for the mechanical advantage, or Reset to clear it.

The bargain behind every machine

No machine gives something for nothing. What a simple machine does is trade force for distance. Ignoring friction, the work you put in equals the work that comes out, and work is force times distance. So if a machine lets you push with less force, it makes you push through a longer distance to do the same job, and the two changes balance exactly.

That is the whole idea of mechanical advantage. A ramp lets you raise a heavy load with a gentle push, but you have to push it all the way up the long slope rather than lift it straight up the short height. A lever lets you lift a rock with a light press, but your end of the lever sweeps through a much bigger arc than the rock rises. The mechanical advantage is how many times the force is multiplied, and it is also how many times further you must move the effort compared with the load.

The formula for each machine

Each machine has its own way of setting the ratio of effort distance to load distance, and that ratio is the ideal mechanical advantage. The calculator uses these:

• Lever: the effort arm divided by the load arm, the two distances from the fulcrum.
• Wheel and axle: the radius of the wheel divided by the radius of the axle.
• Inclined ramp: the slope length divided by the height, which is the same as one divided by the sine of the slope angle.
• Wedge: the length of the slope divided by the thickness of the wide end.
• Screw: the circumference, π times the diameter, divided by the lead, the distance the screw advances in one turn.
• Pulley: the number of rope strands actually supporting the load, which for a standard block and tackle works out to about twice the number of movable pulleys.

In every case a larger number means more force multiplication, bought with more distance.

Ideal advantage and the real thing

What these formulas give is the ideal mechanical advantage, the figure for a perfect machine with no friction. It is set purely by the geometry, the lengths and radii and angles, and it tells you the best the machine could possibly do.

A real machine always falls a little short, because some of your effort goes into overcoming friction between the moving parts rather than moving the load. So the actual force multiplication you measure is always a bit less than the ideal number here, and the gap between them is what the machine's efficiency describes. The ideal figure is still the right starting point: it tells you what the design offers before friction takes its cut.

Units and precision

Mechanical advantage is a pure ratio, a number with no units, since it compares a force to a force or a distance to a distance. That means the units you enter the dimensions in do not matter, as long as you use the same unit for both lengths in a pair, the two lever arms or the ramp's height and length. The result is shown to a few decimal places, which is finer than the dimensions of a real machine are usually known.

A worked example: a lever

Suppose a lever has an effort arm of 1.5 metres from the fulcrum to where you push, and a load arm of 0.3 metres from the fulcrum to the load.

The mechanical advantage is 1.5 ÷ 0.3 = 5. So you can lift a load with one-fifth of the force its weight would otherwise demand, at the cost of moving your end of the lever five times as far as the load rises. For comparison, a ramp 5 metres long and 2 metres high gives 5 ÷ 2 = 2.5, the same result its 24-degree slope gives through one divided by the sine of the angle.

Questions people ask

What is mechanical advantage?

It is the factor by which a machine multiplies your force. A mechanical advantage of 4 means you can move a load with one-quarter of the force you would need without the machine.

Does a machine give you something for nothing?

No. It trades force for distance. Less force always means moving through a greater distance to do the same work, so the total effort is unchanged in an ideal machine.

What is the mechanical advantage of a ramp?

It is the slope length divided by the height, equal to one divided by the sine of the slope angle. A longer, gentler ramp has a higher mechanical advantage, needing less force over more distance.

Why is the real machine weaker than the number says?

Because these formulas give the ideal, frictionless value. A real machine loses some effort to friction, so its actual force multiplication is a little lower, and the difference shows up as less than perfect efficiency.

References

A quick note on where the physics comes from. The six simple machines, the trade of force for distance, and the ideal mechanical advantage formulas are standard mechanics, set out in OpenStax's Physics and described in the Encyclopaedia Britannica. The idea that mechanical advantage is a dimensionless ratio of forces follows the conventions of the US National Institute of Standards and Technology.

1. OpenStax, Physics, Section 9.3, Simple Machines. https://openstax.org/books/physics/pages/9-3-simple-machines
2. Encyclopaedia Britannica, Simple machine. https://www.britannica.com/technology/simple-machine
3. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811