Law of the Lever: Definition, Formula & Examples

Instructor: Matthew Bergstresser
One of the six simple machines is the lever. Levers allow a relatively smaller force to move a heavier object. In this lesson, we will investigate the law of the lever and work some examples involving levers.

Uses of the Lever

Have you ever been on a seesaw or opened a paint can's lid? If so, you were using a lever, which is a bar or other rigid object that has a point to pivot around called a fulcrum.

Let's say you are on a seesaw with your friend who weighs a lot more than you and you want the seesaw to be balanced. You could position yourself at a certain distance from the fulcrum so your lighter body weight counters your friend's body weight allowing the seesaw to be in equilibrium. This means there is no rotation. Let's see how the math works when dealing with a lever.

Law of the Lever

The cross product of force and distance is torque. The cross product is the mathematical process between two vectors that results in a vector perpendicular to both of the initial vectors. The law of the lever is also known as the law of moments and equates clockwise torques and counterclockwise torques. Equation 1 shows the law of levers.

Equation 1

  • F1 is force 1
  • d1 is the distance from the fulcrum force 1 is applied
  • F2 is force 2
  • d2 is the distance from the fulcrum force 2 is applied

Forces applied at a distance from the fulcrum

Imagine trying to open a paint can's lid with a screwdriver, but not applying the force perpendicularly to the screwdriver. If you applied the force at 1° to the screwdriver, it would take a lot of force to open the lid. Applying the force at 90° to the screwdriver would be the most effective at prying open the lid.

The perpendicular force is most effective at rotating the lever

This means we need a way to adjust the amount of torque based on the angle at which the force is applied to the lever. The sine trigonometric function has a maximum value at 90° and a minimum value at 0°. We can rewrite the cross product of force and distance from the lever arm including the sine function. This is how the cross product is evaluated. Equation 2 shows the equation for the magnitude of torques in the law of levers.

Equation 2

  • θ1 is the angle between F1 and the lever arm
  • θ2 is the angle between F2 and the lever arm

We will deal with situations where the angle between the force and the lever arm is 90° so we can remove the sine functions because the sin of 90° is 1. This gives us Equation 3.

Equation 3

Now, we can work some practice problems using the law of levers. Let's get to it!

Example 1

Prompt: Imagine Sal weighs 150 pounds and his friend, Joe weighs 100 pounds. Sal sits 3 feet from the fulcrum on a seesaw. How far does Joe need to sit on the other side of the fulcrum so the seesaw is in equilibrium?

Solution: The first step in solving this problem is to draw a diagram, which we'll call Diagram 1.

Step 1: Draw a diagram

Now we can use Equation 3 to determine how far Joe needs to sit from the fulcrum so there is no rotation of the board they are sitting on.

F S is the weight of Sal, F J is the weight of Joe, d S is the distance Sal is from the fulcurum, and d J is how far Joe has to be from the fulcrum to put the seesaw in equilibrium

Now we can divide both sides by 100 lbs to get the distance Joe needs to sit from the fulcrum to put the seesaw in equilibrium. We'll call this Step 3.

Step 3

So if Joe sits anywhere besides 4.5 feet from the fulcrum, the board they are sitting on will rotate. Let's do another example.

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