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Provided by the Academic Center for Excellence 1 Free-Body Diagrams
Free-Body Diagrams
Introduction
A Free-Body Diagram is a basic two or three-dimensional representation of an object used to show all present forces and moments. The purpose of the diagram is to deconstruct or simplify a given problem by conveying only necessary information. Students may use this diagram as a reference for setting up calculations to find unknown variables such as force directions, force magnitudes, or moments. Free-Body Diagrams allow students to clearly visualize a problem in its entirety or closely analyze a portion of a more complex problem.
Drawing a Free-Body Diagram
In a Free-Body Diagram, the object is represented by its simplest expression, usually a line, box, or dot. The force vectors acting upon the object are represented by straight arrows ( ) while moments are represented by curved arrows ( ) around their respective axis as shown in the diagram below where a force acts at B and a moment around A. The force vectors indicate the magnitude and direction of each force that is acting upon the object. The direction is often indicated by degrees from the horizontal or vertical axis while the magnitude is indicated by units of force. In the case of an unknown magnitude or direction of a force, the unknown value must be labeled as such.
FB
MA
A B
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In addition, it is common to indicate various types of forces with letters and distinguish between common ones by using subscripts. In the example on page 3, weight and tension are represented by W and T respectively, and the force of friction and the normal force are represented by Ffrict and Fnorm respectively. There are no hard rules about how forces are labeled as long as the meaning is clear.
Free-Body Diagrams must also have a labeled coordinate system and include all given dimensions, such as length and angles. Generally, an xy-coordinate system will be used; however, when dealing with a problem in three-dimensional space, an xyz-coordinate system is required. Coordinate systems may be placed according to the student’s discretion in order to simplify the solving process, so long as the student follows the right-hand rule (see Example Step 5 on page 5).
Types of Forces
In order to be able to identify and label forces for a Free-Body Diagram, students must recognize the various types of forces they will encounter and know how the forces interact with each other in order to calculate them.
Weight: Any object with a mass has a weight. It can be given in pounds or newtons (N). If the weight is not given, it can be calculated in newtons by multiplying the mass in kilograms with the Earth’s gravitational constant (9.8 m/s^2 ).
Normal Force: As explained by Newton’s Third Law, every action has an equal and opposite reaction. Due to this law, any object that is not in free fall (i.e. an object that rests on a surface) has a normal force acting upon it perpendicular to the surface the object is resting on. Absent of any additional forces, the magnitude of the vertical component of the normal force is equal to the weight of the object.
Provided by the Academic Center for Excellence 4 Free-Body Diagrams
Setting up the Free-Body Diagram
Step 1: Draw the object with no extraneous features
Step 2: Identify the forces present. The box has mass, so it also has weight , a force acting downward. Because the box is on a surface, there is a normal force acting perpendicular to the surface. Attached to the box, there is a rope with tension applied. This force will act in the direction of the rope. Since the rope is attached to the box, in order to move it up the incline, there will be a frictional force impeding movement. This force will act in the opposite direction, down the incline.
Step 3: Add the forces to the drawing of the object and label the directions of the forces in degrees from the vertical or horizontal axis as determined by the geometry in the example. W Fnorm
Ffrict
T
W Fnorm T Ffrict
30°
60° 30°
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Step 4: Label all known values. At this point, all that is known is the weight, which is the mass (50 kg) multiplied by the gravitational constant (9.8 m/s^2 ). The Free-Body Diagram now contains all the given, important information.
Step 5: As a general rule, the Free-Body Diagram should be oriented, so the direction of movement is along one of the principle axes. In this example, the entire diagram can be reoriented by rotating it 30° counter-clockwise. This step results in the direction of movement occurring along the x-axis, and it results in three of the four forces also being oriented along the x or y axis.
Note: When dimensions are provided in the initial problem, it is essential to label all dimensions in the Free-Body Diagram.
Solving the Free-Body Diagram
In order to solve the problem, the force on the rope necessary to move the box up the incline must be found. This is the tension force. Finding this force requires a system of equations. Although there is currently one known variable, the weight , there are three unknown variables; therefore, three equations are required. These equations establish the relationship between each of the forces and are necessary in order to solve for each force.
W = 50 kg • 9.8 m/s^2 Fnorm = 490 N
Ffrict
T
60° 30°
30°
W = 490 N
x
y
60°
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Now that the system of equations is found, the unknown variables can be determined by inputting known information, namely that the weight is 490 N. Equations are then combined in order to solve for tension.
Fnorm = sin(60°) • 490 N = 424.4 N
424.4 N • 0.25 = Ffrict = 106.1 N
106.1 N + cos(60°) • 490 = T = 351.1 N
With tension found, the answer to the problem is at least 351.1 N of force must be applied to the rope in order to move the box up the incline.
When working with Free-Body Diagrams involving torque, please refer to the Academic Center for Excellence’s handout, Moments and Torques.
