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Matthew has a Master of Arts degree in Physics Education. He has taught high school chemistry and physics for 14 years.

The free-body diagram is a powerful way to analyze forces in a scenario. This lesson describes the rules you should follow when using a free-body diagram, and includes multiple examples modeling how to solve force problems.

The Visual Component

Newton's second law of motion is one of the most famous physics equations along with Einstein's E = mc2. Even though Î£F = ma may look like a simple equation, it's actually quite sophisticated because of the sigma in front of the F. This sigma means that all forces acting on an object must be added together in each respective direction independently. Keeping track of the multitude of forces that are acting on an object is the challenge when solving problems related to force and acceleration. A free-body diagram is a way to visually inventory these forces, and helps with developing the algebraic equations needed to solve these types of problems.

There are five rules for drawing free-body diagrams:

To represent the object the forces are acting upon, draw either a dot or a box. It's frowned upon to depict the actual object. For example, if the object is a car, resist the temptation to draw a sketch of the car. The point of the free-body diagram is to keep any scenario, as Einstein supposedly mused, 'as simple as possible, but not simpler.'

Only include forces in the diagram, and represent them with arrows. No velocity vectors, acceleration vectors, or any other vectors should be drawn on the dot or box.

The length of the arrows should represent the relative magnitude of the force. They don't have to be to scale, though.

Include a coordinate system with one of the axes parallel to the direction of the acceleration.

All force vectors (arrows) should be pure forces, not component forces. In other words, don't use trigonometric functions in the free-body diagram.

Example 1

Let's go through a few examples involving the use of a free-body diagram.

Analyze the free-body diagram and determine the net force on the mass.

Since force is a vector, we have to isolate each force based on the axis to which it's parallel. In the y-direction, there's a 6 N force acting in the positive direction and a 6 N force acting in the negative direction. These forces cancel, so there's no acceleration in the vertical direction. In the x-direction, there's a 20 N force acting to the right and a 3 N force acting to the left. The difference between these two forces is 17 N acting to the right; therefore, the object is accelerating to the right.

Example 2

A mass is on an inclined plane that's Î¸ = 53o above the horizontal. A 20 N force is pulling the mass up the incline while a 10 N force (gravity) is pulling it down to the earth. Determine the acceleration of the mass.

Notice the coordinate system has the x-axis parallel to the inclined plane.

The 10 N force acting down is the weight of the mass due to Earth's gravity. Since the acceleration due to Earth's gravity (g) is known, we can determine the value of the mass:

To keep the free-body diagram pure, we'll create a new component-force diagram analyzing the component forces acting on the mass, both in the y-direction and the x-direction.

The red arrow is the component of the weight perpendicular to the incline, and the green arrow is the component of the weight acting down the incline.

We know the angle Î¸ and the weight; therefore, we can determine the value of the red side of the triangle, the side adjacent to Î¸.

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Since the component of the weight is the same as the normal force, we can conclude that there is no acceleration perpendicular to the incline.

Now we can sum the forces in the x-direction to determine if there's a net force on this mass, and therefore an acceleration in the x-direction.

Since there is an unbalanced force pointing up the ramp, the mass accelerates up the ramp.

Example 3

Mass M is equal to 2m, and a 20 N force accelerates both masses at 2.5 m/s 2. What force accelerates mass M?

When dealing with multiple masses in a system, each mass must be treated independently and have its own free-body diagram. This is always where we should start when solving force problems.

Now we can set up the algebraic equations we'll use to solve the problem. There is an unknown force between the two blocks (NsubH). We can solve for this force, which pushes back against the 20 N force, by knowing that the acceleration at the end is 2.5 m/s 2.

Lesson Summary

Newton's second law of motion is very powerful, and the use of free-body diagrams is essential to inventory all of the forces acting on a mass. Drawing a free-body diagram is the first thing that should be done when dealing with a force problem. It helps when dealing with the Î£F side of the equation.

The five rules for drawing a free-body diagram are:

Represent the mass with a box or dot.

Only include forces in the free-body diagram, and represent them with arrows pointing in the direction of each force.

The length of the force vectors (arrows) should represent the relative magnitude of the forces.

Include a coordinate system with one axis parallel to the direction of the possible acceleration.

Don't include component forces in the diagram. If necessary, you can draw a separate diagram that deals with the component forces.

The free-body diagram helps visualize the forces acting on a mass, which makes the setup of the algebraic equations easier. Since force is a vector, the forces in each respective direction must be treated independently of each other. If the net force in either direction is not zero, there is an acceleration in that direction.

When dealing with multiple masses in a system, be sure to draw a free-body diagram for each mass. Often there's at least one common force acting between two masses. The use of algebraic substitution will then be required to solve the problem.

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