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AP Physics 1: Exam Prep13 chapters | 143 lessons | 6 flashcard sets

Instructor:
*Matthew Bergstresser*

A free-body diagram is a visual way to represent the forces acting on a mass. This lesson will go through how to set up and solve the mathematical expressions required to determine the acceleration of the mass.

Drawing a free-body diagram is the first step in determining the acceleration of a mass using Newton's second law: Σ F = *ma*. Sometimes, a problem will provide the free-body diagram for you, and your job is to write the mathematical expressions including the forces from the diagram to solve for the acceleration.

The first thing to consider before writing mathematical expressions from a free-body diagram is how many dimensions are involved? Each axis must have its own mathematical expression because force is a vector. **Vectors** have a specific direction associated with them, and only like directions can be summed.

Let's start with an example where the free-body diagram (FBD) is provided and we have to turn the information in the diagram into a mathematical expression.

Prompt: Use the FBD to determine two mathematical expressions for *a*, one in the x-direction, and one in the y-direction. Then describe how to use the x and y accelerations to get the overall (net) acceleration.

Solution: In our example there are only 2 axes therefore 2 dimensions.

Any forces that are not parallel to a pure direction (parallel to the x-axis, y-axis, or z-axis) must be multiplied by a trigonometric function of its related angle. This will get the magnitude of the force in the pure directions, x and y. Now we are ready to turn FBD 1 into a mathematical expression.

Now we can determine the x and y accelerations. Let's use algebra to get each expression solved for the accelerations.

To get the net acceleration we have to set up a right triangle and use the Pythagorean theorem. Since this example doesn't include numbers, let's say that its x-acceleration is positive, and its y-acceleration is negative.

Now, let's do an example where we use numbers to determine an acceleration.

Prompt: Use the FBD to determine the acceleration of the 5 kg mass.

Solution: First we break the vectors up into the respective directions, and then solve for the acceleration in each direction, using 5 kg for *m*.

The mass accelerates in the positive y-direction at 1 m/s2 .

Physics requires a lot of practice, so we will do another example.

Prompt: The FBD shown is for a 10 kg mass sliding up a ramp ( θ = 30o ) and friction is acting on the mass. The friction force is equal to μ = 0.4 times the normal force, and the normal force is mgcosθ . Determine the acceleration of the mass.

- f is the friction force
- N is the normal force
- M is the mass
- g is the acceleration due to gravity ( 9.8 m/s2 )

Solution: Notice that the x-y coordinate system is tilted. This is commonly done when a mass is sliding along an inclined surface. The purpose is to get as many forces in the scenario parallel to both the x and y axes as possible. It allows us to determine a minimum number of component forces. The normal force is in the direction of +y, and the friction force is in the -x direction. The weight of the mass is the only force not parallel to an axis. If we didn't tilt the axes, we would have both the normal force and the friction force that would need to be broken down into component forces like we did in Example 1.

Notice how we substituted μ N for friction force (f), and then substituted Mgcosθ for the normal force. Even though each axis must be treated independently, sometimes there are variables common to both directions, and substitution is required.

**Free-body diagrams** are visual ways to represent the forces acting on a mass. These diagrams are helpful to solve for any variable in the scenario including the acceleration, *a*. Newton's second law is ΣF = ma, and the Σ means we have to sum all of the forces on the mass, but since force is a vector each direction must be treated independently. A **vector** is an entity that has a magnitude and a direction, therefore only forces in the same line can be summed.

For each coordinate axis where forces are acting, one Σ F = ma equation must be used. For example, if there are forces acting in the x and y directions, two Σ F = ma equations must be used; one for each direction.

The forces in each direction are plugged into the left side of Newton's second law, and set equal to the mass times the acceleration (ma) in each respective direction. Algebra can be then used to solve for acceleration in each direction. If there is an acceleration in each direction, the Pythagorean theorem can be used to determine the net acceleration.

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AP Physics 1: Exam Prep13 chapters | 143 lessons | 6 flashcard sets

- Newton's First Law of Motion: Examples of the Effect of Force on Motion 8:25
- Distinguishing Between Inertia and Mass 6:45
- Mass and Weight: Differences and Calculations 5:44
- State of Motion and Velocity 4:40
- Force: Definition and Types 7:02
- Forces: Balanced and Unbalanced 5:50
- Free-Body Diagrams 4:34
- Solving Mathematical Representations of Free-Body Diagrams
- Net Force: Definition and Calculations 6:16
- Force & Motion: Physics Lab
- Newton's Second Law of Motion: The Relationship Between Force and Acceleration 8:04
- Determining the Acceleration of an Object 8:35
- Action and Reaction Forces: Law & Examples 8:15
- Determining the Individual Forces Acting Upon an Object 5:41
- Implications of Mechanics on Objects 6:53
- Air Resistance and Free Fall 8:27
- Newton's Third Law of Motion: Examples of the Relationship Between Two Forces 4:24
- Newton's Laws and Weight, Mass & Gravity 8:14
- Identifying Action and Reaction Force Pairs 8:12
- The Normal Force: Definition and Examples 6:21
- Friction: Definition and Types 4:15
- Inclined Planes in Physics: Definition, Facts, and Examples 6:56
- Go to AP Physics 1: Newton's Laws

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