How to Calculate Levers & Leverage

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Levers allow you to apply an input force at one point of the lever to create an different output force at a another point on the lever. The leverage, which is the output force divided by the input force, may be greater or less than one, depending on your needs. The key feature of a lever is the fulcrum, a point on the lever held stationary while other parts move. First-class levers have the fulcrum between the input and output forces. Second- and third-class levers have the fulcrum at one end. With second-class levers, the input force is at the far end and the output force closer to the fulcrum; with third-class levers, the reverse is true.

First-Class Lever: Scissors

    Draw two lines crossing at a small angle to represent the blades of a scissors. Label a small distance from the pivot, or fulcrum, as L_paper = 1.5 centimeters.

    Label a larger distance from the fulcrum as L_handle = 12 centimeters.

    Use the formula F_handle_L_handle = F_paper_L_paper to calculate the leverage, or mechanical advantage, which is F_paper/F_handel = L_handel/L_paper = 8.

    Calculate the distance you need to move the handle to have the blades cross by 2 millimeters at the contact point with the paper using 2 millimeters_L_handel/L_paper = 2 millimeters_12/1.5 = 1.6 centimeters. The goal is to have the scissor blades apply a large force on the paper over a small distance. To do this, you apply a smaller force but need to move the handles a larger distance.

Second-Class Lever: Wrench

    Draw a wrench turning a nut. Label the distance from the center of the nut, which is the fulcrum, to the outside edge of the nut as L_nut = 1 centimeter. Label the distance from the center of the nut to the end of the wrench as L_wrench = 20 centimeters.

    Draw an arrow at end of the wrench and label it as F_wrench. Assume it will take 1,500 Newtons to loosen the nut, equivalent to 337.2 pounds. Draw an arrow at the outside of the nut and label it as F_nut = 1,500 Newtons.

    Calculate F_wrench using the formula F_wrench_L_wrench = F_nut_L_nut to get F_wrench = F_nut*L_nut/L_wrench = 75 Newtons, equivalent to 16.9 pounds. You need to apply 16.9 pounds of force to start turning the nut.

    Calculate the leverage as F_nut/F_wrench = 1,500/75 = 20.

Third-Class Lever: Elbow Joint

    Draw a simplified bent elbow with the upper arm bone -- the humerus -- meeting the forearm bone -- the ulna -- at 90 degrees. The joint is the fulcrum.

    Draw the bicep muscle from the shoulder end of the humerus to the ulna at a point close to the joint. Label the distance from the joint to the bicep L_bicep = 2 inches. Label the distance from the joint to the end of the ulna, where the hand would be, as L_hand = 16 inches.

    Draw an arrow at the hand and label it as F_hand = 10 pounds of force. Draw an arrow along the bicep and label it as L_bicep.

    Calculate F_bicep using the formula F_bicep_L_bicep = F_hand_L_hand to get F_bicep = F_hand*L_hand/L_bicep = 80 pounds. The bicep applies a force of 80 pounds at the elbow to allow the hand to apply a force of 10 pounds.

    Calculate the leverage as F_hand/F_bicep = 0.125. The goal is to have the elbow joint only move a little while the hand moves a lot. This requires a mechanical advantage of less than one.


About the Author

Ariel Balter started out writing, editing and typesetting, changed gears for a stint in the building trades, then returned to school and earned a PhD in physics. Since that time, Balter has been a professional scientist and teacher. He has a vast area of expertise including cooking, organic gardening, green living, green building trades and many areas of science and technology.

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