Free Body Diagrams.

A Free Body Diagram is a way of analysing a physics problem by breaking it down to a simpler arrangement. This is particularly helpful when dealing with a complex device with many parts and connections.Almost every engineering problem begins with a Free Body Diagram.

Action / Reaction

All it starts with Newton’s three Laws of motion. Well. Only the third one actually, and we aren’t even talking about motion just now. But it still applies.

Every action has an equal and opposite reaction.

Take an object sitting on a table. A wooden block.

The block pushes down on the table. At the same time, the table also pushes UP on the block. This is what Newton meant – every action (another word for force) has its counterpart.

For now we will only consider situations where nothing is accelerating (which include things that are stationary of course). Every point of contact where force is applied will produce a pair of equal-and-opposite forces. So every contact force has a counterpart.

Lets keep going. The ball pushes down on the table, and the table pushes up on the ball. An equal and opposite pair. It’s common sense really. If the table didn’t push up on the ball, the ball would accelerate downwards.

Any other points of contact?Yes, the contact between the table and the floor. The table pushes down on the floor, and the floor pushes up on the table.

Now, we haven’t drawn any gravity forces yet. These are always taken to act at the centre of an object’s mass, called the centre of mass. Logical.

Gravity is down.Always.

If we could isolate the floor to see what it is sitting on we could keep on going all the way to planet earth itself. But we won’t because we have to stop somewhere. That’s an engineering trick right there – knowing when to stop. The discipline of knowing when to stop is the essence of the Free Body Diagram.

OK, so if we include all the forces involved with the ball and the table we get this;

This seems rather complicated for such a simple problem, doesn’t it? Fortunately many forces cancel each other out.

Let’s say the ball has a gravity force of 50 Newtons (50N). So F causes C to push DOWN at 50N, which means E pushes UP at 50N. (Newton’s third law of equal and opposite).Now add C and E together and we get ZERO! (in other words, C and E cancel each other out)

This always happens at the counteracting force pairs

If we keep going, the forces at every contact point will cancel out. But we can’t keep going here, we don’t know enough about the floor – it is sitting on two walls… three piers… directly on the ground? No idea. So we are stuck.

The Free Body Diagram.

The Free Body Diagram puts a limit on which body (or bodies) the engineer chooses to include. The diagram above includes everything. We call this a space diagram. Now we need to do a Free Body Diagram (FBD).

Free: This means we will select the part (or group of parts) we are interested in, and cut it FREE from everything else.

Body: This is the chosen part (or group of parts) that we want to study. It must have forces we know and forces we are trying to find.

Once the body is selected, the most important task is to choose which force in each counter-part pair is right one.

Here’s a simple rule. Include only those forces that are applied TO THE BODY. The easiest way to identify these is to label each force like this;

For example. Force E = What does the table do TO the ball

A FBD of the ball means the body is the ball. So we include only those forces that are applied TO THE BALL;

Every other force is irrelevant. So are the other bodies, like the table and the floor. They are still having an effect because the ball is not falling, but they are taken out of the diagram and replaced with their forces (only the ones that act on the body).

In fact, engineers will draw the FBD in a simplified form without showing any other bodies. It is a simplified representation like the one below. After all, it is called a Free Body Diagram, not a Free Body Picture.

FBD for Ball

Here is the basic method for making a FBD;

  1. Isolate the body you want
  2. Find connection points where force pairs exist
  3. Select the correct force from each force pair according to the rule: TO THE BODY

Example: FBD for Table.

  1. Isolate the body you want (TABLE)
  2. Find connection points where force pairs exist (TABLE GRAVITY, BALL CONTACT, FLOOR CONTACTS)
  3. Select the correct force from each force pair according to the rule: TO THE TABLE

And this is how an engineer would draw the FBD of the table.

Originally we had 8 forces, but by choosing the body as the table only 4 forces are relevant..

If the engineer was trying to design the table and calculate stresses, they MUST start with a FBD like this BEFORE any engineering analysis can be done.

How to Draw a Free Body Diagram

1. Isolate Body

There must be only one body.Make sure you isolate the body exactly – as if you cut it out in a silhouette or outline. It is important to be very clear about the boundary you have made around thebody.

2. Find Force locations

Forces are applied by contact, gravity or inertia. Identify all the points where forces are applied to the body. Gravity always acts through the centre of mass, pressure acts through the centre of pressure.

The only forcesto consider are those thatCROSS THE BOUNDARY. Ignore all other forces – those inside the boundary, and those forces that belong to outside bodies that do not contact the boundary.

3. Line of Action of Forces

  • Point contact: ‘Smooth’:No friction. Force can only be applied perpendicular to smooth surface.
  • Wheels:Perpendicular to the rolling surface and acting through centre of axle.
  • Cable. Force can only be tensile (pulling). Force must be along direction of cable.
  • Pin Joint: Force can be any direction but no moment (torque) around pin.
  • Other Joints: Solid:Force in any direction, moment in any direction.
  • Slider:Moment (torque) in any direction, but force must be perpendicular to slider
  • Gravitational. Centre of gravity. Always down (270o)
  • Inertial. Centre of inertia (not always the same as C.O.G). These are due to acceleration.
  • Fluid pressures. Usually taken as a single resultant force in statics calculations, acting through the centre of pressure.Fluid is frictionless, so can only act perpendicular to surface.
  • Other. Magnetic, electrostatic (charge) etc. These are pretty rare outside of electric equipment design.

4. Direction of the Forces

Include all the forces actingTO THE BODY. The direction is determined by thinkingTO the body, orON the body.Eg Gravity acts down ON the body, floor pushes up ON the body, what does the road do TO the wheel…etc.

Making a Free Body Diagram (Short Version)

  1. Isolate the body.
  2. Locate contact points (border crossings).Don’t forget gravity.
  3. Line of Action.Back in theSpace Diagram, determine the line-of-action of forces using the rules for standard connections (support reactions). Now find the single point of intersection for the three forces. This will give all the angles.
  4. "To the Body".Identify the direction as applied "to the body".

Support Reactions

In step 3 of the FBD, you may need to determine the angle of some forces. (Called the line of action)


FromVector Mechanics for Engineers:Beer, Johnston, Eisenberg,Staab p165

Worked Example 1

A loaded wheelbarrow being held up on level ground. (We are pretending it is two dimensional for simplicity. This is OK because each hand carries an equal load – half each)

There are many bodies here. A hand, wheel, dirt, handle, legs, bolts, tray, axle, tyre, ground…

We need to select an appropriate body according to what forces we know (e.g. weight of loaded wheelbarrow), and what forces we want to find out (e.g. force at the hand).

A body that has ALL of these is the ENTIRE wheelbarrow.

This is s group of parts – taken as a single body.

  1. Isolate the body.
  2. Locate contact points (border crossings).Don’t forget gravity.
  3. Line of Action.Back in theSpace Diagram, determine the line-of-action of forces using the rules for standard connections (support reactions). Now find the single point of intersection for the three forces. This will give all the angles.
  4. "To the Body".Identify the direction as applied "to the body".

Now for Step 2: Contact Points. These have to be at the BOUNDARY of the body, not within it. Gravity is the only force we include that come from within a body. This gravity force must be the weight of the entire wheelbarrow + load.

There are three points to consider: hand, gravity and wheel contact.

  1. Isolate the body.
  2. Locate contact points (border crossings).Don’t forget gravity.
  3. Line of Action.Back in theSpace Diagram, determine the line-of-action of forces using the rules for standard connections (support reactions). Now find the single point of intersection for the three forces. This will give all the angles.
  4. "To the Body".Identify the direction as applied "to the body".

Now we are up to Step 3: Line of Action…

Direction of forces for the three contact points: hand, gravity and wheel contact.

  • Gravity is vertical down.
  • Wheel is the next easiest. This is a roller, so force is perpendicular to the surface (vertical).
  • Hand is a type of cable or short link, so it pulls. Since the first 2 forces are vertical, this last one must also be vertical)

Gravity force C. Downwards.

Hand contact point: A equal and opposite to B.

Wheel contact point: D equal and opposite to E.

  1. What the wheelbarrow does to the hand
  2. What the hand does to the wheelbarrow
  3. What gravity does to the wheelbarrow
  4. What the ground does to the wheelbarrow
  5. What the wheelbarrow does to the ground
  1. Isolate the body.
  2. Locate contact points (border crossings).Don’t forget gravity.
  3. Line of Action.Back in theSpace Diagram, determine the line-of-action of forces using the rules for standard connections (support reactions). Now find the single point of intersection for the three forces. This will give all the angles.
  4. "To the Body".Identify the direction as applied "to the body".

One to go.Step 4.TO THE BODY…

We need to select those forces according to the format:

Since our body is the Wheelbarrow, we will only include the forces applied TO THE WHEELBARROW.

  1. What the wheelbarrow does to the hand
  2. What the hand does to the wheelbarrow
  3. What gravity does to the wheelbarrow
  4. What the ground does to the wheelbarrow
  5. What the wheelbarrow does to the ground

Which gives a FBD for the wheelbarrow.

An engineer would use this FBD to work out the force at the hand grip, or the maximum weight the barrow can hold, or the wheel load. Almost every engineering problem begins with a Free Body Diagram.

Worked Example 2

Find the forces ina frame that supports a certain load.

We start with a given load and need to find the forces in the strut and column.

There are 3 main bodies (not counting nuts and bolts) – Beam, Column and Strut.

What do we use for a body?

  • The Strut? FBD is possible, but since there is no known force connected to it, there are no numbers to start with. Can’t start here.
  • The Column? No. This body is incomplete, but the real problem is we have no known forces on it.Can’t start here either.
  • The Beam? Yes. This has input information (LOAD) so this is a good place to start.
  • The Beam and Strut? Yes, this could work too. Not much different to using the Beam for a body.

Free Body Diagram for Beam