New Mexico Science Olympiad
Mission Possible

Last updated 11/8/07

Calculating Mechanical Advantage

Mechanical advantage is the result of using simple machines to reduce the force required to move a load, or to increase the distance a load is moved. It can be thought of as "gain" where a smaller force is converted to a larger force by increasing the distance it is acted over, or a smaller distance is converted to a larger distance by increasing the force that is applied over the shorter distance.

Moving a load from one place to another requires a fixed amount of work. Work is defined as Force times distance moved.

W = F x D

In this equation we see that as distance increases, force decreases. Typically when machines are used to reduce the force required to move a load, it is accomplished by increasing the distance. For example, to elevate a load from one elevation to another with the minimum distance traveled you must lift it vertically. This straight vertical lift requires the maximum force as the distance is minimized. Using an inclined plane to change the elevation increases the distance the load moves to reach the desired elevation, thus reducing the force required. This reduction of force is referred to as mechanical advantage. We can think of it like this where A = the mechanical advantage:

W = (1/A)F x AD

We have basically multiplied the right side of the equation by A/A which equals one: Force has decreased and distance has increased, but they are scaled to each other so work is preserved.

If we use no mechanical advantage the force is not reduced (A=1), so we call this a mechanical advantage of 1, or "unity gain". (This is important to note that "no mechanical advantage" is not where A=0, it's where A=1 at unity gain where force in equals force out) If we double the distance thereby reducing the force by half (A=2), we call this a mechanical advantage of 2.

Here are specific procedures to calculate your mechanical advantage:


A lever can be used to reduce force to move a load a particular distance or to move a load a larger distance for the particular force. This can be measured in several ways, all of which are ultimately a function of the ratio of lever arm segment lengths (the equatios for this depend on the type of lever you're using). Typically the mechanical advantage is calculated as a ratio of forces, where A = mechanical advantage:

A = (force of load) / (force of effort)

It can also be measured as a ratio of displacements:

A = (displacement of load) / (displacement of effort)

Inclined Plane

With the inclined plane you are doing two things. You're increasing the distance the load travels to do the same work (ignoring friction). You're also dividing the force on the object into vertical and horizontal components. We're going to exhibit a horizontal force on an object that is moving diagonally upwards, thereby creating a force vector. A force vector has both a horizontal and vertical component. We will use the following formula, where A = mechanical advantage:

A = (length of inclined surface) / (height of incline)

This is a fraction derived from the formula 1 / (sin θ) where θ = angle of incline.

By trig identities we know that (sin θ) = (height of incline) / (length of inclined surface)

So 1 / (sin θ) inverts that fraction and we end up with the formula above.

Note that a straight vertical lift makes this fraction = 1 which denotes a 1-to-1 relationship between force on object and vertical load, therefore where there's "no mechanical advantage" then the result is 1. Keep in mind that the length of inclined surface is the hypotenuse of the triangle created (the length of the actual surface the load travels on), it is not the length of the base of the triangle.

Pulley System

In order for a pulley system to reduce the force required to lift a load (or in a horizontal example perhaps to compress or expand a spring), the length of pull on the string must be increased in order to reduce the force required to pull it. With pulleys this can be accomplished by multiple passes through multiple-sheaved pulleys. This will increase the length of pull required to move the load a fixed distance. Therefore, the simplest way to determine mechanical advantage is in a ratio of length of pull to the displacement of the load:

A = (length of pull) / (displacement of load)

If your load needs to move 10 cm and you set up your pulley system so that it pulls 30 cm of string to move the load 10 cm, then you have a mechanical advantage of 3. If you simply use a single pulley where the length of pull is equal to the displacement of the load, you have a mechaical advantage of one, which is considered no advantage.

This can also be measured as a ratio of the force of pull on the string and the force exerted on the object, but in all likelihood this is going to be impossible to measure in your machine on event day, so be sure we can determine your mechanical advantage using the pull length/displacement formula.

For the 2007 competition the rules essentially require a mechanical advantage greater than 1 for the lighter can to lift the heavier can. To calculate the minimum mechanical advantage for this particular case, use the labeled values on the cans for the weight, then simply make a ratio of the load to effort:

m = (load) / (effort) = (weight of lifted can) / (weight of lifting can)


Good luck and Have Fun!

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