Video Transcript
In this video, we’re talking about
free body diagrams. As we’ll see, these are diagrams
that show us the forces that act on a particular isolated object. Free body diagrams help us
understand how an object will or will not move under the influence of forces. And they’re especially useful in
scenarios with multiple interconnected masses. One of the best ways to start
learning about free-body diagrams is to see them in action. Let’s say we have this classic
physics scenario of a block sliding down an incline plane. To understand the motion of this
block, it will be helpful to learn about its acceleration.
Now, in and of itself, we may not
know anything about how this object accelerates. But if we refer to Newton’s second
law of motion, this law tells us that the acceleration an object undergoes is
related to the forces applied to it. So if we can learn about the forces
on this block, we’ll know something about its acceleration. And this is where free-body
diagrams get involved. We’ve already been specific about
which object we want to understand the motion of. It’s this block. That is, it’s not the incline the
block is sliding on or the block in inclined system, but just the block itself. So the first step in creating our
free-body diagram of the forces acting on this block will be to isolate it, to draw
it all by itself.
So here’s our block drawn by
itself. Now, what we want to do is start
counting up the forces that are acting on this block. Starting off that list, we know for
one thing that gravity is a force on the block. And then partly counteracting that
force is the normal force, sometimes also called the contact force, between the
incline in the block. And then, let’s assume our incline
plane is not smooth, but instead is rough. That is, it applies a frictional
force to the block. At this point, we’ve named all the
forces that are active on the block, gravity, the normal force, and the frictional
force. Our next step is to draw these
forces in on our diagram of our block using arrows.
Now, the reason we use arrows that
will have a certain length in a certain direction is because these three forces
we’ve named, just like all forces, are vectors. That is, they point a certain
way. And they have a particular
magnitude or value in that direction. Now, when we draw a free-body
diagram, one way to do it, is to pick a point at the centre of the object of
interest and draw in all the forces as though they come from that point. Taking this approach, the force of
gravity would be drawn in something like this, an arrow starting at that point and
going downward. The normal force would be normal or
perpendicular to the angle of the incline. That is, it will point in this
direction. And the frictional force, like
always points in the direction opposing the object’s motion. So that would be up the incline
like this.
Notice that, just like we expected,
not only do these arrows point in different directions representing the direction of
the force, but they also have different lengths representing different force
magnitudes. We can see, for example, that the
arrow representing the force of gravity is longer than the arrow representing the
frictional force. This means that, relative to the
frictional force, gravity is stronger. It has a greater magnitude. Now, before we go further, we
should label these arrows. Otherwise, we might get the forces
they represent confused. The arrow pointing downward is the
gravitational force; we’ll call that 𝐹 sub 𝑔. The arrow pointing up into the left
is the normal force; we’ll call it 𝐹 sub 𝑁. And then up into the right is the
frictional force; we’ll call that 𝐹 sub 𝑓.
As we look at these three forces,
we can see that they’re not all at right angles or perpendicular to one another. In particular, the angle between
the force of gravity and the normal force and the force of gravity and the
frictional force is not 90 degrees. Now, this is a bit of an issue
because, remember, we want to be able to figure out what is the net force acting on
this object and use that to figure out how it may be accelerating. But it will be difficult to solve
for the net force on our block if the individual forces acting on it are hard to
combine. But there’s a step we can take to
make this easier. What we can do is define a
coordinate axes, 𝑥𝑦-directions, on our free-body diagram.
For example, we could say that the
positive 𝑦-direction of these axes that we’re defining points the same way as the
normal force. And then we could say the positive
𝑥-direction points in the direction of the frictional force. And even though it doesn’t
completely look like it, these axes are at right angles to one another. With those drawn in, we can see
that the normal force and the frictional force are entirely either along the 𝑦- or
the 𝑥-axis of our graph. But the gravitational force is an
exception. It’s split between these two. Part of it is in the 𝑥-direction,
and part of it is in the 𝑦. But what we would like to do is to
break up or divide this gravitational force into the components in the 𝑥-direction
and the 𝑦-direction.
Here’s how we can do that. We can say that this force vector,
the gravitational force arrow, is the hypotenuse of a right triangle where the two
other sides of this triangle are the 𝑥 and 𝑦 components of this force. So if the overall gravitational
force is represented by this arrow as we’ve drawn it, then the 𝑦 component of that
force is represented by this one and the 𝑥 component by this one. By dividing the gravitational force
into the directions that we’ve called 𝑦 and 𝑥, respectively, we’ve made it easier
to compare this force with the other ones acting on our free body. By doing that, we make it much
simpler to solve for the net force acting on this block, which then tells us a bit
about its acceleration.
Now, what we’ve drawn here is a
very specific free-body diagram for this particular block sliding down this
particular plane. But the steps we followed in
drawing up this diagram are ones that can apply to any object. Off to the side then, let’s write
in just what were those steps that we followed. The first thing we did was we
isolated our object of interest, in our case, the block sliding down the plane. Part of doing this involved in
drawing a simple sketch of that block. Now, this block was already pretty
simple, but in some cases we may see a free-body diagram that’s even more
simplified. It’s possible to represent an
object, a mass, by a dot, for example. We didn’t do it exactly that
way. But there is some variation in the
free-body diagram generation process.
Once we had our simplified sketch,
though, we moved on to step two. And that was to list the forces
that we’re acting on our object. It’s important to specify that
they’re on the object and not forces the object exerts on something else and recall
that, in our case, those forces were gravity, the normal force or contact force, and
the frictional force. Once all the forces were accounted
for, our next step was to draw them in as vectors, that is, arrows on our
diagram. Those arrows pointed at certain
direction, depending on the direction of the force. And their length indicated the
relative strength of that force. Next, we saw that it was important
to put labels on the heads of these arrows. So we could tell the forces
apart.
Since all the arrows represent
forces, the only way these labels deferred was in their subscript, 𝐹 sub 𝑔
compared to 𝐹 sub 𝑁 compared to 𝐹 sub 𝑓. With these steps complete, we moved
on to our final one. And that is to draw in coordinate
axes by our free-body diagram. That way, we’ll know which
direction is positive 𝑦 and which is positive 𝑥. With this four-step process, we
understand how to draw a free-body diagram. And as we saw in the case of our
block, this is all motivated by a desire to understand the motion of the object of
interest. When we understand the forces on
that object, and in particular the net force, we can better understand how it may or
may not be in motion. Let’s get a bit of practice now
with these ideas through an example.
A box is pulled along a surface
by an applied force of 32 newtons as shown in the diagram, which is not to
scale. The net horizontal force on the
box, acting to the right is 24 newtons. What is the magnitude of the
force 𝐹?
Okay, so looking at our
diagram, we see our box right here on top of this surface in blue. Our statement tells us that the
box is pulled along the surface by an applied force of 32 newtons we see
there. And it’s the force acting in
the other direction 𝐹 whose magnitude we want to solve for. Now, even though we’re not told
it explicitly, what we have in this diagram is a free-body diagram of our
box. That is, we have a depiction of
all the forces acting on this object drawn in as vectors. So we see, for example, that
there’s this 16-newton force acting down. We can guess that that’s the
weight force or the gravitational force on the box. And then there’s this equal and
opposite 60-newton force acting up. That’s likely the normal or
contact force. And then, as we saw before, the
32-newton force is an applied force on the box. And the force 𝐹, of course, is
our unknown that we want to solve for.
In order to solve for 𝐹, we’ll
need to account for all the forces we see in this free-body diagram. But there’s something very
helpful about these forces that makes our task simpler. And that is that two of these
forces, the 16-newton force acting up and the one acting down, are perpendicular
at right angles to the force 𝐹. That means they have no
influence on forces in that direction, in the horizontal plane left or
right. Even if they did, by the way,
the fact that these two forces are equal in magnitude but opposite in direction
means that they effectively cancel one another out. 16 newtons of force pushing up
and 16 newtons of force pushing down add up to a net force of zero. So the only forces we need to
keep in mind are the 32-newton force and the one we want to solve for, 𝐹.
We can start solving for 𝐹 by
using some information given in the statement. We’re told that the net
horizontal force on the box acts to the right and has a magnitude of 24
newtons. That means that if we combine
the force 𝐹 acting to the left and the 32-newton force acting to the right,
then together they yield 24 newtons acting to the right. Now, before we go further,
let’s decide on a horizontal direction that we’ll call positive. Let’s say that motion to the
right is in the positive direction. And that means that motion to
the left is in the negative. Using this convention, if we
add together all the forces in the horizontal direction, here’s what we
find.
First, there’s our 32-newton
force, and that’s a positive value because it’s acting to the right. Then we subtract from that
force the force 𝐹, the one we want to solve for. The reason for this minus sign
is 𝐹 points to the left, which we’ve decided is in the negative direction. We’re told that when we add
these two forces together, the sum or the net is 24 newtons acting to the right
and, therefore, a positive number. This is the equation we’ll use
to solve for our unknown force 𝐹.
To do that, let’s begin by
adding 𝐹 to both sides of the equation. When we take that step, we can
see we have a plus 𝐹 and a minus 𝐹 on the left-hand side. So they cancel one another
out. And then following that, let’s
subtract 24 newtons from both sides, which, we can see, results in 24 newtons
minus 24 newtons adding up to zero on the right-hand side. And that leaves us with this
equation. 32 newtons minus 24 newtons is
equal to the force 𝐹. And that, we can tell, is eight
newtons. That’s our final answer. Eight newtons is the magnitude
of the force 𝐹.
Let’s now summarise what we’ve
learned about free-body diagrams. Starting off, we saw that a
free-body diagram, sometimes abbreviated FBD, shows all the forces that act on a
particular object. Free-body diagrams we saw are
useful for understanding the net force on an object and, therefore, its motion. And finally, we learned a four-step
process for drawing free-body diagrams. Step one is to isolate the object
of interest. Step two is to list all the forces
acting on that object. These may include forces like
gravity, tension, normal or contact force, and so on. Step three is to draw those forces
in on our isolated object. We use arrows where the arrow
direction indicates force direction and the length of the arrow indicates the
relative force strength. Once the arrows are drawn in, we
label them with the name of the force they represent. And lastly, we pick and then draw
in a pair of coordinate axes. So we know which way is positive 𝑥
and which way is positive 𝑦. This is the process for drawing a
free-body diagram.
