# Undergraduate Mathematics/Free body diagram

Figure 1: Block on a ramp (top) and corresponding free body diagram of just the block (bottom). For equilibrium, the line of action of the three force arrows must intersect at a common point.

A free body diagram, sometimes called a force diagram,[1] is a pictorial device, often a rough working sketch, used by engineers and physicists to analyze the forces and moments acting on a body. The body itself may consist of multiple components, an automobile for example, or just a part of a component, a short section of a beam for example, anything in fact that may be considered to act as a single body, if only briefly. A whole series of such diagrams may be necessary to analyze forces in a complex problem. The free body in a free body diagram is not free of constraints, it is just that the constraints have been replaced by arrows representing the forces and moments they generate.

## PurposeEdit

Drawing a free body diagram can help determine the unknown forces on, moments applied to, and equations of motion of, the body and thus help to analyse a problem in statics or dynamics. In analysis of structures, free body diagrams for a component of a structure or, part thereof, are used in determining shear forces and bending moments.[2][3]

## ConstructionEdit

A free body diagram, or FBD, is not meant to be a scaled drawing. Rather it is a working sketch open to modification as one works through the problem and typically one needs to have seen through the problem before one arrives at a satisfactory diagram. There is an element of art, an inherent flexibility in the whole process. There is no hard and fast algorithm. The iconography of a free body diagram - not only how it is drawn but also how it is interpreted - depends crucially on how a body is modelled.[4]

### Modelling the bodyEdit

A body may be modelled in three ways:

(i) a particle. This model may be used when any turning effects are zero or have zero interest even though the body itself may be extended. The body may be represented by a small symbolic blob and the diagram reduces to a set of concurrent arrows. A force on a particle is a bound vector.

(ii) rigid extended. Stresses and strains are of no interest but turning effects are. A force arrow should lie along the line of force, but where along the line is irrelevant. A force on an extended rigid body is a sliding vector.

(iii) non-rigid extended. The point of application of a force becomes crucial and has to be indicated on the diagram. A force on a non-rigid body is a bound vector. Some engineers use the tail of the arrow to indicate the point of application. Others use the tip.

### Example: a body in free fallEdit

The foregoing remarks can be elucidated with the help of an example. Consider a body in free fall in a uniform gravitational field. The body may be

(i) a particle. It is enough to show a single vertically downward pointing arrow attached to a blob.

(ii) rigid extended. A single arrow suffices to represent the weight W even though gravitational attraction acts on every particle of the body. The arrow must lie along a line through the centre of gravity. Where exactly along the line the arrow should lie is a matter of convenience.[4] A popular choice is to locate the arrow at the centre of gravity but strictly speaking there is no need to do so in rigid body analysis. See figure 2.

Figure 2: An empty rigid bucket in free fall in a uniform gravitational field with the force arrow at the centre of gravity

(iii) non-rigid extended. In non-rigid analysis, it would be a positive error to associate a single point of application with the gravitational force.

### What is includedEdit

An FBD represents the body of interest and the external forces on it.

1. The body: This is usually sketched in a schematic way depending on the body - particle/extended, rigid/non-rigid - and on what questions are to be answered. Thus if rotation of the body and torque is in consideration, an indication of size and shape of the body is needed. For example The brake dive of a motorcycle cannot be found from a single point, and a sketch with finite dimensions is required.
2. The external forces: These are indicated by labelled arrows. In a fully solved problem, a force arrow is capable of indicating
(i) the direction and the line of action[notes 1]
(ii) the magnitude
(iii) the point of application.

Typically, however, a provisional free body sketch is drawn before all these things are known. After all, the whole point of the diagram is to help to determine these things! Thus when a force arrow is originally drawn its length may not be meant to indicate the unknown magnitude. Its line may not correspond to the exact line of action. Even its direction may turn out to wrong. Very often the original direction of the arrow may be directly opposite to the true direction. An engineer may also omit some forces altogether, especially in rigid body analysis where there are paired forces which cancel each other.

The forces acting on the object include friction, gravity, normal force, drag, tension, or a human force due to pushing or pulling. When in a non-inertial reference frame, fictitious forces, such as centrifugal pseudoforce may be appropriate.

A coordinate system is sometimes included, according to convenience. This may make defining the vectors simpler when writing the equations of motion. The x direction might be chosen to point down the ramp in an inclined plane problem, for example. In that case the friction force only has an x component, and the normal force only has a y component. The force of gravity will still have components in both the x and y direction: mgsin(θ) in the x and mgcos(θ) in the y, where θ is the angle between the ramp and the horizontal.

### What is excludedEdit

Although there is nothing to stop the engineer from having supplementary sketches to help elucidate the problem situation, the free body diagram proper - an FBD - should not show

1. bodies other than the free body.
2. forces exerted by the free body.
3. internal force exerted by one part of the free body on another part. For example, if an entire truss is being analyzed to find the reaction forces at the supports, the forces between the individual truss members are not included.

Any velocity or acceleration is left out. These may be indicated instead on a companion diagram, called a "Kinetic diagram", "Inertial response diagram", or the equivalent, depending on the author.

A remark is in order concerning the second point 2. above. By Newton's 3rd law if a body A exerts a force on a body B then B exerts an equal and opposite force on A. This equality of two opposite forces which act on two different bodies is often confused for quite a different equality which pertains to a body in equilibrium subject to two equal and opposite forces. A diagram showing the forces exerted on and by a body is likely to be misconstrued as showing a body in equilibrium and is best avoided.

## Example: A block on an inclined planeEdit

A simple free body diagram, shown above, of a block on a ramp illustrates this.

• All external supports and structures have been replaced by the forces they generate. These include:
• mg: the product of the mass of the block and the constant of gravitation acceleration: its weight.
• N: the normal force of the ramp.
• Ff: the friction force of the ramp.
• The force vectors show direction and point of application and are labeled with their magnitude.
• It contains a coordinate system that can be used when describing the vectors.

Some care is needed in interpreting the diagram. The line of action of the normal force has been shown to be at the midpoint of the base but its true location can only be found if sufficient further data is given. The diagram as it stands would need to be modified were we told that the block is in equilibrium.

There is a potential difficulty also with the arrow representing friction. The engineer who drew this diagram has used the tip of the arrow to indicate the point of application of a force. (See the other force arrows in the diagram). Now, the tip of the friction arrow is at the highest point of the base. The intention however is not to indicate that the friction acts at that point. The engineer in this instance has assumed a rigid body scenario and that the friction force is a sliding vector and thus the point of application is not relevant. The engineer has tried to indicate that the friction acts all along the whole base by drawing an arrow all along the base but such artistic ploys are a matter of personal choice.