In physics we deal with gravity, normal forces, friction, drag, and the list goes on. Sometimes we even deal with all of these forces at the same time. How can we keep track of all of these different pieces? Free-body diagrams are one tool that physicists use to make sense of all of these different and sometimes opposing forces in order to understand how the system will react.
Meaning of Free-Body Diagrams
A free-body diagram allows us to keep track of all of the forces acting on an object.
A free-body diagram is a type of vector diagram that displays an object and the forces acting on it.
To draw a free-body diagram, draw on arrows extending from the center of mass of the body to show the forces that act on it. The size of the arrow tells us the magnitude, and the direction the arrow points in is the direction of our force. This can show us if our forces are unbalanced so that our system is accelerating, or if we have a static problem, where we have a net force of zero. Such a diagram is an indispensable tool for many types of physics problems. Note that when we initially draw our diagram, we may not know the magnitudes of the forces. Typically an estimate is sufficient to keep track of what forces are acting on our object. In the course of solving for the values numerically, we may find that our initial force magnitudes were not quite correct. The object that the forces are acting on is usually represented by a point or a simplified drawing of the object. Everything else is removed to make it clear what object we are interested in.
Types of Forces Commonly Displayed on a Free-Body Diagram
Some of the more common forces seen on a free-body diagram are the force of gravity, which pulls an object towards the Earth, and the normal force which pushes an object away from the point of contact. Friction and drag work against the direction of motion to slow a moving object. The force of tension works to pull objects. The centripetal force keeps an object from flying away in rotational motion, and always points toward the center of the circle of rotation.
It is important to understand what forces should be displayed, and what forces should not be displayed. The only forces shown on a free-body diagram are the forces acting on the object. Any forces exerted by our object should not be displayed. For example, if we consider a box sitting on the floor, we should include the normal force of the floor pushing up on our box, but not the normal force of our box pushing down on the floor.
Example of extraneous forces drawn on a free-body diagram, StudySmarter Originals
Here the force of gravity is given as \(F_\text{g}\), the normal force in the first image, \(F_\text{g}\), is the normal force of the floor acting on the box. This is the correct way to display a free-body diagram. The second image on the left shows \(F_{\text{N}_1}\), the normal force of the floor acting on the box. It also incorrectly shows \(F_{\text{N}_2}\), the normal force of the box acting on the floor. This is incorrect: we should never show forces exerted by our object, but only forces acting on our object.
Your free-body diagram should only include one object. If you wish to include more, you must draw a separate diagram. This is why it is called "free": we are looking at one isolated object.
Free-Body Diagram Equations
The basic premise of a free-body diagram is to qualify all the forces. This suggests we use Newton's second law,
\[F_\text{net}=ma.\]
This is a good place to start, but there are some subtleties we need to consider. For example, is the net force on our object equal to zero? If so, then our forces should be balanced, but if not, our force vectors should not all cancel out. Considering formulas is a good way to start a problem, but don't forget to also consider the underlying physics that the formulas are describing.
Examples of Free-Body Diagrams
Now that we understand a bit more about what a free-body diagram is, let's take a look at some examples.
Stacked Blocks
A common example that shows a somewhat more complicated use of normal forces is a stack of blocks.
Consider the case of three stacked concrete blocks. Draw free-body diagrams for each block.
Stacked blocks for a free-body diagram problem, StudySmarter Originals
Remember that we need to draw a free-body diagram for each block separately. The top block, \(A\), is the simplest, so we will start there. All we have acting on this block is the force of gravity, which we will call \(F_\text{g}\), and the normal force supplied by block \(B\) to block \(A\). We will call this \(F_{\text{N}_{BA}}\).
Free-body diagram of block \(A\), StudySmarter Originals
Note that this is a static system so the sum of the forces on each block should be zero, as well as the net force on the system. In this case, the normal force should cancel the force of gravity.
Next, we look at block \(B\). This is a little more complicated. First, we have our force of gravity, \(F_\text{g}\), which is stronger because block \(B\) is larger. We also have two normal forces. The normal force from block \(A\) pushing down on block \(B\), denoted \(F_{\text{N}_{AB}}\), and the normal force from block \(C\) pushing up on block \(B\), denoted \(F_{\text{N}_{CB}}\).
Free-body diagram of block \(B\), StudySmarter Originals
Finally, we have block \(C\). This looks similar to block \(B\), but the only difficulty is keeping the normal forces straight, and making sure the magnitudes of all of our forces make sense. As usual, we have our force of gravity, \(F_\text{g}\). Our normal forces are the normal force from \(B\) pushing down on \(C\), \(F_{\text{N}_{BC}}\), and the normal force of the ground pushing up on \(C\), which we denote as \(F_\text{N}\).
Free-body diagram of block \(C\), StudySmarter Originals
Blocks Connected Over Pulley
Pulleys redirect forces making them a slightly more interesting case than a standard free-body diagram.
Draw a free-body diagram for each block connected over the pulley. Note that the blocks are made of the same material.
Weights on a pulley - StudySmarter Originals
When we draw our free-body diagram, we keep in mind that the force of tension is equal everywhere in a pulley system. So we have the diagram below.
Free-body diagram of block \(A\) in the pulley system, StudySmarter Originals
Here, \(F_{\text{g}_A}\) is our force of gravity, and \(F_\text{T}\) is our tension. Similarly, we have the following.
Free-body diagram of block B in the pulley system, StudySmarter Originals
Here again, \(F_{\text{g}_A}\) is our force of gravity and \(F_\text{T}\) is our tension.
Friction in a Free-Body Diagram
We can also examine what friction forces would look like in a free-body diagram.
Consider a block sliding down a ramp with a coefficient of kinetic friction \(\mu_\text{k}\). Draw a free-body diagram for the block.
Problem setup of a block on a ramp, StudySmarter Originals
The forces we are dealing with are the normal force, gravity, and friction. Gravity points downwards, the normal force is perpendicular to the ramp, and friction acts against our direction of motion.
Free-body diagram of a block on a ramp with friction, StudySmarter Originals
To break these into components, we can split up the normal force and the force of friction, however; there is a trick to make our lives a little easier. If we rotate our coordinate axes to line up with the normal force, we only have to split gravity into its components.
Free-body diagram with rotated axes, StudySmarter Originals
Now we rotate the whole system so it is easier for us to see. We are not changing anything in the diagrams below, we are just changing our perspective so that \(y\) points up again.
Rotating the system to return \(y\) to the up direction, StudySmarter Originals
Lastly, we can split up our force of gravity and we are done.
Free-body diagram with rotated axes and all forces split into their components, StudySmarter Originals
Although not strictly necessary, it is usually helpful to deconstruct forces into their \(x\) and \(y\) components. Problems often require looking at the components of forces, not just the forces themselves.
Centripetal Force Free-Body Diagram
For our last example, we will consider how to draw a free-body diagram when we have uniform circular motion.
Consider a ball attached to a rope on a pole that is swinging around it in a uniform circular motion. Draw a free-body diagram for the ball.
Ball attached by a rope to a pole undergoing uniform circular motion, StudySmarter Originals
Note that the only two forces acting on the ball are the force of tension, \(F_\text{T}\), and the force of gravity, \(F_\text{g}\), so our free-body diagram is relatively simple.
Free-body diagram for ball connected to a pole by a rope, StudySmarter Originals
So where does the centripetal force come in? In this case, the centripetal force is the net force on our object. We can see what the net force is if we break up our forces into their \(x\) and \(y\) components. Gravity is already acting in only the \(y\) direction, so we just need to break up the tension.
Free-body diagram with tension split into its components, StudySmarter Originals
The ball is not moving up and down, so our forces in the \(y\) direction cancel out. Now it is easy to see that our centripetal force is the force of tension in the \(x\) direction. Note that as the ball spins around, this force will always point toward the center of the circle of rotation. This confirms that it is our centripetal force.
Free Body Diagrams - Key takeaways
- Free-body diagrams show all the forces acting on an object clearly and concisely.
- Free-body diagrams show both the magnitude and direction of the forces.
- You never include the forces that the object imparts on other objects, only the ones that act on our object.
- For each object, you draw a separate free-body diagram.
- Decomposing the forces into their \(x\) and \(y\) components is often necessary to solve problems.
- When splitting forces into their \(x\) and \(y\) components, rotating the \(x\) and \(y\) axes can sometimes reduce the number of forces we need to decompose and make our lives easier.
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