Let’s talk about equilibrium of rigid bodies. Imagine we have a flower pot hung up like this.
There is a weight to this flower pot which is straight down. So what’s keeping it still,
in other words, in equilibrium? It’s the rope that’s attached to it, which is creating an
opposite force, in other words, a tension in the rope that’s holding it in place.
Now imagine we have an object like this, and there are a bunch of forces and moments effecting it.
If we sum all the forces and moments, meaning we find the resultant force and the couple moment,
and they are both equal to zero, then that object is in equilibrium. In simple terms, the object is
not moving. So this equation means that the sum of all the forces acting on the object is equal to
zero. We can break this equation down to forces in the x direction and y direction. So all the forces
in the x direction added together must equal zero and the same for y axis forces. This equation
means the sum of all the moments, including ones being applied, and ones being created
by the forces, is equal to zero. Today, we are going to look at just two-dimensional problems,
so just objects in the x and y plane. When we look at an object, we need to look at something called
support reactions. These reactions at the supports are the forces and couple moments that keep an
object from moving. They create opposite forces and moments so that an object doesn’t translate
or rotate. So in our flower pot example, the rope is a support reaction. The rope creates a tension
which keeps the flower pot from falling. Let’s explore more support reactions. Here we have a
bar. there are 2 support reactions. One is a roller and the other is a pin. Let’s focus on
just the roller first. Assume that our object can move, and we apply a force from the left,
notice how our bar moves. That’s because a roller can only prevent vertical movement. If we apply
a force from top, the roller prevents the bar from moving down by creating an opposite force.
So if we’re given an object with a roller as a support, that roller can only exert a vertical
force which prevents the object from translating in the vertical direction. Now, let’s look at the
pin side. With a pin, the bar can rotate, but it can’t move vertically or horizontally. So that
means at a pin, it would exert 2 forces, one in the x direction and another in the y direction.
In all of supports we just looked at, rotation can occur, however, imagine we fix our beam to
the ground or to a wall. Now it can’t move in the horizontal direction or the vertical direction,
and it also cannot rotate. In this case, we’d have 2 reactions, one in the x axis and another in the
y axis, and we would also have a couple moment because rotation cannot occur. In other words,
an opposite couple moment must be developed so that the bar doesn’t rotate due to the moments
created by forces. Your textbooks will contain a list of these supports and what type of reactions
these supports develop, so please take a look at those. Lastly, to solve the problems you face,
you must know how to break forces into components and how to figure out moments created by forces.
If you don’t remember, or need a refresh, please check the description.
Now let’s get to some examples and see how we can actually solve for unknown reactions.
Let’s take a look at this problem where we need to find the reactions at pin A and the
tension in the rope. The first step is to draw a free body diagram. Since we have a pin at A,
we know there will be 2 reactions, Ax and Ay. Next, we look at the forces in our diagram.
We have a 26kN force, and we draw it broken into it’s x and y components.
Then we have a 40 kN force straight down. Lastly, we have the tension in the rope, which can also be
broken into it’s x and y components. It is time for us to write our equations of equilibrium.
First, we will consider the x-axis forces. What we are saying is that for this object to stay in
equilibrium, the sum of all the forces in the x direction must equal to zero. We will assume
right to be positive. So we have Ax, then we have the 10kN force, but notice how our 10kN force is
negative, that’s because we assumed right to be positive and our force is facing left. Lastly, we
have the x component of the tension in the rope. All of this added together is equal to zero. Next,
an equation for y axis forces. We will assume up to be positive. It’s the same as before, but
now we are just looking at the vertical forces. We now have 2 equations with 3 unknowns, which
means we need one more equation, which is going to be a moment equation. This is usually the case
with most questions, you will need 3 equations, 1 for x axis forces, 1 for y axis forces, and then
one moment equation. We want to eliminate as many unknowns as possible, so we are going to write our
moment equation about point A, which gets rid of Ax and Ay. Remember, where ever we calculate the
moment about, we ignore the forces applied at that point since their line of action will go through
that point. So for example, if we picked point B, then we ignore the forces at point B. Now we
will pick clockwise to be positive and write our moment equation about point A. The 10kN force,
which is our x component won’t create a moment since it’s line of action goes through point A,
the same goes for the x component of the tension in the cable. So only the y components create
moments. With this, we now have 3 equations with 3 unknowns, so you can solve them anyway you like.
Solving gives us the reactions at pin A and the tension in the cable. Notice that
we got a negative value for AX, all it means is that its opposite to the direction we assumed.
Let’s take a look at this problem where we need to find the reactions at the roller A
and pin B. Our first step is to draw a free body diagram. Since it’s a roller at A,
it will only have a vertical reaction. At B, we will have both the x and y reactions.
The distributed load has to be expressed as a resultant force, which can be found
by multiplying the load by the length, giving us 12 kN, and it will be placed at the center,
so 2 m away from point B. Now we can write our equations of equilibrium.
First, the x-axis forces. Note that the reaction at A is at an angle of 30 degrees,
so we can break it into x and y components. Next, the y-axis forces.
We see that we have 2 equations with 3 unknowns, so we need one more equation,
which will be a moment equation about point B. We pick point B because there are 2 unknowns there,
and we always want to get rid of as many unknowns as possible. We will assume clockwise to be
positive. Focusing on the force at A, we have 2 components, both with create a moment. For the
x component, we have a perpendicular distance of 3sin 30 degrees. For the y component, we have 4m
plus 3 cos 30 degrees. The only other force is the 12 kN force, which is 2 m away from point B.
We now have 3 equations with 3 unknowns. Solving gives us the reactions at A and B.
In this question, we need to figure out the reactions at A and at B. Let’s draw a free
body diagram. Since it’s a smooth collar at A, we will have a force perpendicular to the rod,
and a moment created about A. At point B, which is a smooth contacting surface, we will have a
reaction force that acts perpendicular to the surface. Since the forces are perpendicular,
all that happens is that the given ratio triangles just flip. There is also a 100
N force that’s straight down, and we have a moment of 20 nm applied at the bend. Now we
are going to write our equations of equilibrium. Let’s start with an equation for x-axis forces.
Note that we are using the x components of the forces.
Next, an equation for y-axis forces. From these two equations,
we can solve for forces FA and FB. Now we still need to figure out the
moment created about the smooth collar. For that, we can write a moment equation about point A and
we will assume clockwise to be positive. So at A, the reactions at A are ignored since their
line of action goes through point A, however, we still need to consider the moment about point A.
That moment must be included in our equation. Next, we have the 100N force,
and the perpendicular distance is 0.3 m. After that, we have the 20 Nm applied at the bend, it’s
a clockwise moment, so it’ll be positive. Finally, we have the x and y components of our force at B.
For the x component, the perpendicular distance is 0.2 m and for the y component, it’s 0.6 m.
The x component creates a clockwise moment and the y component creates a counter clockwise moment.
Solving gives us the moment at A, but notice that it’s a negative value.
All it means is that the moment is opposite to our assumption, so it’s actually counter clockwise.
Let’s take a look at one last example. In this example, we have a weight attached to a rod
and on one end, it’s pinned, and at the other end, we have a spring. We need to figure out the angle
created when the system is in equilibrium. First, we have to draw a free body diagram.
At pin A, we have 2 reactions. We also have the moment applied at A. Note that this moment is
not created about pin A by other forces, it is something that was manually applied. Now
we look at the weight of the cylinder, that’s mass times the acceleration due to gravity.
Here, I think it requires a bit of intuition, but the angle that’s created at this point, is
the same angle that’s created here because if we place our coordinate system like this and move it,
you can see that it’d be the same. Lastly, we have the force of the spring that’s straight upwards.
Now we have to put on our thinking caps, so initially, the spring has an unstretched length
of 1 m. In the second state, when everything moved, the spring stretched a bit. How much did it
stretch? Well, let’s draw a right-angle triangle like this. Notice how the length of stretch is
the opposite side. So we can use sine to figure it out. Now that’s the length the spring stretched,
so to find the force, we use hooks law. The stiffness is 600 N/m. Let’s simplify.
Now also notice that when we draw this force, the angle that it creates is once again the same.
Now we can figure out this angle, in fact, we actually only need a single moment equation
about point A. That would get rid of the 2 unknown reactions, and only leave the angle to be found
since we know all the other forces. Let’s go over this equation. First, we have the 600 Nm
moment applied at A. Then we look at the weight of the cylinder. Only the y component of the weight
creates a moment since the x component’s line of action goes through point A. The perpendicular
distance is 1.5 m. The same is true for the force of the spring. Only the y component
does any work and the perpendicular distance is 3 m. Note that we assumed clockwise moments to
be positive so the spring force creates a counter clockwise moment. Now we can solve for theta. We
are looking for values between 0 degrees and 90 degrees. If you graph it, you will see that there
are 2 such values, and those are our answers. Always remember that if an object is in
equilibrium, the sum of the forces must equal zero. With that, I hope this video helped
and if it did, please consider sharing it with your friends and class mates,
they too might find it helpful. Thanks for watching and best of luck with your studies!