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Understanding Parallelogram of Forces

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

To find the resultant of two forces, you can use the parallelogram of forces law. In this lesson, learn about why the parallelogram of forces works and how to apply it.

Why Does the Ball Roll Down the Hill?

Sammie is kicking a ball and she accidentally kicks it towards a steep hill. As soon as the ball reaches the hill, it starts to roll down and keeps going faster and faster the farther down the hill it rolls. Looking at the ball, Sammie wonders why the ball speeds up as it rolls down the hill. This is a problem that intrigued some great thinkers throughout history, like Isaac Newton.

Proving the Parallelogram Law

To understand what is going on with the ball, you have to consider the forces that act on it. While the ball is rolling down the hill, it is acted on by two forces: a downward force of gravity and a force from the surface of the hill that we call a normal force. The normal force is always perpendicular to the surface, so it is exerted at an angle.


Forces on an inclined plane


How can we add these two forces together to find out what will happen to the ball? This was something that Isaac Newton wondered as well. He said to first consider what would happen if each force was applied independently. The force of gravity would cause the ball to speed up in a downward direction since it is directed downward, and the normal force would cause the ball to speed up in the direction that it is acting (upward at an angle). This would cause the ball to move in two different directions at the same time. Combining both of those types of motion, the ball would move down the slope.


parallelogram law proof


Based on his reasoning about the combined velocity of the ball, Newton said that you could combine these two forces into just one force, called the resultant force that would have the same effect on the ball as the two forces did individually. To find the resultant, you would make a parallelogram with the sides equal to the two applied forces. The diagonal of this parallelogram will then be equal to the resultant force. This is called the parallelogram of forces law.

Let's see how you could apply it to the ball rolling down the hill.


parallelogram law definition and example


Adding the two forces applied to the ball together gives you a resultant force that points straight down the hill. Because the resultant force pushes the ball down the hill, it will move in that direction and speed up as it goes.

Applying the Parallelogram of Forces Law

Now that you know how the parallelogram of forces law works, let's try to apply it to calculate the resultant force in a couple of common situations. First, what is the resultant of a 10 N force directed upward and a 5 N force directed to the right?

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