Free Body Diagram

by Whitney Graham

Edited by pgupta66@gatech.edu (Fall 2016)

Main Idea

What are Free Body Diagrams?

A free body diagram, or force diagram, is a rough sketch that shows the relative magnitude of all the forces acting on a system. There are various forces that can be acting on the object, such as applied force, frictional force, normal force, and gravitational force. A free body diagram allows for analysis in a steady state condition, where there is no acceleration on a system. All forces in a free body diagram are due to the system's interactions with its surroundings. Especially when problems become complicated and involve different forces acting on multiple objects, free body diagrams can be extremely effective in making a problem simpler to handle.

Mathematical Modeling

Since a free body diagram is a snapshot of a steady state condition, there is no net acceleration when we are modeling a free body diagram. Therefore, we can use the following equations to conclude:

Fnet = Mass x Acceleration

Since acceleration is equal to 0:

Fnet = 0

Additionally, if acceleration is equal to zero, then

a = dv/dt --> there is constant velocity during snapshot of the diagram

Relation to Newton's Second and Third Law

Free body diagrams are usually used in collaboration with Newton's Second Law, F=mass*acceleration, as both are typically used in the process of solving for force. Newton's Second Law is the sum of the forces. The sum of the forces is equal to zero when the object is not accelerating or is moving at constant velocity. Additionally, since there is no net force acting on the body during a free body diagram, we can use Newton's Third Law of every force has an equal and opposite force to ensure that in our free body diagram all the components of the force in the x and y direction equal to zero during calculation.

How To Draw a Free Body Diagram

(1) Establish a Coordinate System

Before beginning to analyze a system, it is important to choose an appropriate coordinate system (an x and y direction plane) that will be the most convenient to avoid dealing with complicated angles of forces and difficult arithmetic. This is especially true when dealing with angled forces where tilting the x and y plane could make solving for resulting forces a lot easier. Also, in order to maintain consistency with direction amongst various people solving the same system, an x-y coordinate establishment can ensure that anyone solving the same system will end up with the same directions on their forces they are solving for.

This is an example of how establishing a coordinate system can make solving/identifying forces easier

(2) Identify the system

Often times, you will be faced with single and multi particle systems where bodies within that system will be interacting with/exerting forces on each other. For example, a block is resting on top of a larger block, and both are moving down a ramp. In that case, analyzing the top block versus the bottom block will result in different free body diagrams. Therefore, identifying which part of the system you are analyzing before beginning to draw a free body diagram will allow for a more efficient solving process.

(3) List and identify all surroundings that interact with the system

Although we usually think of these interactions in terms of force names, it's best to get in the habit of identifying the force AND the object that specifically causes that force. For example, the force of gravity on a block could be caused by the earth. Or a normal force could be caused by a second block and another normal force could be caused by the table. Identifying forces by the specific objects that cause them helps us not forget forces.

Types of Forces to Consider for Free Body Diagrams:

Disclaimer: Not all of these forces will be present in every situation. These are not all possible choices of forces, just the most common in a Physics 1 course.

Applied Force: Force applied to the system by a person or other object.
Force of Friction: Force that a surface applies on the system that is moving (or trying to move) on that surface.
- Formula: f=μN
- f = Frictional Force
- N = Normal Force
- μ = Coefficient of Friction
Force of Gravity: Force that, on Earth, will act downward toward the center of the Earth.
- Formula: Force of gravity = mg OR -(GMm)/r^2
- m = mass
- g = 9.8 m/s^2 (on earth)
Normal force: Force that is present when the system is on another object or surface, and the object or surface is exerting a force on the system as support.
Spring force: Force that is exerted by a spring onto any system that is attached to it.
- Formula: Spring force=-kx
- k = spring constant
- x = displacement from the spring's relaxed position
Force of tension: Force that exists when a rope, string, wire, etc. is pulling on the system.

(4) Draw a diagram with the system at the center

Can use a dot to represent the system, OR
Can draw the details of the system (Draw a block, car, etc.)
Make sure to draw the coordinate system next to the system

(5) Draw all the forces acting on the system

(represented as arrows). Arrow length should represent the approximate magnitude of that force relative to other forces

(6) Label all forces with a symbol

representing the name of the force and identify them by the object causing the force
Examples of force symbols:

Fg (force of gravity)
Ff (force of friction)
Ft (force of tension)
Fn (normal force)
Fc (contact force)

Examples of objects causing the force:

Earth
Ramp
Block
Rope
Moon
Spring

(7) Break forces into their components as needed.

If a force is acting diagonal to the system, create a dashed line parallel and perpendicular to the system and label it as the x and y components of that force. Use sine(θ) or cosine(θ) as needed.

General Tips:

If an object has constant speed, it means the object has no acceleration. Since net force is the rate of change of acceleration, net force in that direction would then be zero. This means that there are either no forces currently acting on the object, or there are equal, opposite forces acting on the object in that direction. To represent this in a free body diagram, draw forces as arrows pointing in opposite directions with equal lengths.
Don't be confused by contact forces. Most of the time, contact force is an umbrella term that includes other types of forces. If you had a block on a ramp, you could draw the contact force as being diagonal to the ramp. Or, if you wanted to break it into its components which are easier to consider, you would draw the Normal force caused by the ramp pushing up as a perpendicular arrow, and the force of Friction caused by the ramp as a parallel arrow (see image to the upper right).

In-Depth Tutorial

(Using Steps Outlined Above)
Ex: A block is stationary on a ramp.

Identify the system: block
List all objects interacting with the system:
1. Ramp
2. Earth
Draw a diagram of the system: Draw a block to represent the system.
Draw all forces acting on the system: Because the block is stationary, we know the forces in each direction must sum to equal zero.
Label all forces with a force symbol and identify the object causing the force
1. Normal force, Fn, caused by the ramp
2. Force of friction, Ff, caused by the ramp
3. Force of gravity, Fg, caused by the earth
Break forces into components: We can break up the contact force into normal force and force of friction.

Setup
Step 3
Step 4
Step 5
Step 6

Examples

Simple

Example 1 Solution-In this example, the only forces acting on the system are the force of gravity and the normal force

Example 1: A person rides in an elevator moving at constant velocity. Create a free body diagram to represent this situation.

Middling

Example 2 Solution-In this example, the only forces acting on the system are the force of gravity and the force of tension

Example 2: A ball is hanging on a string of negligible mass from the ceiling. Create a free body diagram to model this situation.

Note: For this example, when solving for forces, the force of tension must be split into horizontal and vertical components and solved.

Difficult

Example 3 Solution-Because the sled is accelerating to the right, net force must be positive in the x direction. To achieve this, the applied force must be greater than the force of friction.

Example 3: A force is applied to the right to accelerate a sled to the right. Draw a free-body diagram.

Note: Friction is necessary to keep the box stationary. When solving for the normal force and for frictional force, both the x and y component must be considered.

Diagrams with solution are on the left and right.

Connectedness

Free body diagrams are the building blocks for many scientists, physicists, engineers and have many applications in the real world. They form the basis for determining a device's or building's structural stability, safety, and overall practicality.

Medical Applications

There have recently been many advances in medicine in terms of wearables and prosthetic limbs that function just like normal body parts for either congenital defects or accident recovery. Let's take a look at a leg prosthetic. While walking, a prosthetic exerts many forces on your leg such as: the force of gravity, normal force from the ground, frictional force, etc. Prosthetics move like springs, while your body moves forward it also moves up and down which causes changes in kinetic energy due to movement and changes in gravitational potential energy. While it contracts and relaxes, different energies are converted back and forth into kinetic and potential energy. In order to analyze all the forces on a prosthetic to ensure it's safety and usability, the most basic step when designing one is a free body diagram.

Buildings and Trusses

Many civil engineers and architects are responsible for building trusses, bridges, and buildings which function using the basic concepts of physics. It is important to ensure that components of a truss and bridge are in equilibrium so that one day they don't collapse. When building a truss bridge, engineers put straight members in place that form the bridge's top and bottom, and they are linked by a structure of diagonals and vertical posts. For trusses especially, all the components must remain in equilibrium, the magnitudes of forces exerted must be equal, and the components in tension and compression must be identified. In order to create a proper structure, a free body diagram must be used by engineers in the preliminary stages for design and safety purposes.