How to calculate levers & leverage

Updated March 23, 2017

Archimedes famously said, "If you give me a lever long enough and a place to stand, I can move the world." A lever is a bar pivoted around a stationary point, called a "fulcrum," to gain mechanical advantage, also known as "leverage." This means that the lever can reduce the force required to lift a load on the opposite side of the fulcrum. For example, pushing down with a force of 22.7 Kilogram on one side of a lever 2 yards long can lift a load weighing 45.4 Kilogram, if the side of the lever on the other side of the fulcrum is 1 yard long.

Draw a diagram of a lever with the fulcrum underneath. You can represent the fulcrum with a triangle, with its top point touching the underside of the lever. Draw the load on the shorter end of the lever, a distance X from the fulcrum's point of contact with the lever. Denote the distance from the fulcrum to the point where you'll input force on the longer end of the lever with the letter Y. Use W to represent the weight of the load. Suppose further that you know that you can push down on the lever with no more than Z pounds of force.

Measure the length of your lever, assigning X and Y to the two ends of the lever, as in Step 1. Denote its length with the letter L. So X+Y = L.

Determine the ratio of the required output force to the input force. This is just W/Z. You'll need this much leverage in order to lift the load W.

Determine the ratio of the input lever arm Y to the output lever arm X that is needed so that the mechanical advantage Z/W can be achieved. Therefore, Y/X = W/Z.

Determine X and Y in terms of the total length, L, of the lever. Combining Y/X = W/Z and X+Y = L gives Y = (W/Z)(L-Y), or Y=WL/(Z+W). Therefore, X=ZL/(Z+W).

Returning to the example mentioned in the introduction, Y = WL/(Z+W) =100x3/(150) = 2 yards. X = ZL/(Z+W) = 50x3/(150) = 1 yard. So the fulcrum needed to be placed such that Y was at least 2 yards long to gain the required minimum leverage of 2-to-1.

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About the Author

Paul Dohrman's academic background is in physics and economics. He has professional experience as an educator, mortgage consultant, and casualty actuary. His interests include development economics, technology-based charities, and angel investing.