In this session, we are going to learn how
to solve particle equilibrium problems.
Let’s start with a simple a problem.
Take a look at this example.
Two workers are trying to place a heavy create
on the floor.
The crate is supported by two ropes.
Let’s assume the mass of the crate is 100
kg.
Rope BC makes an angle alpha with the horizontal.
Rope AB makes an angle beta.
At this instant, this crate is in equilibrium.
Our goal is to determine the forces experienced
by these two cables.
Before we go any further I want to make remind
you something about cables.
Cables, ropes and wires work only on tension.
In other words, I can pull a cable and generate
force on the cable but we cannot push or compress
a cable to support a load.
This is a useful information because when
you see a cable or a rope, you now know the
direction of the force.
The force is always pulling or stretching
the cable.
This makes our life easier in solving problems.
In solving static equilibrium problems, I
want you to adopt a consistent approach.
First you want to identify what is given,
and what you need to find out in the problem.
Second you need to draw a Free Body Diagram
Third you must write the equations of equilibrium.
And that is, sum of all the forces equal to
zero.
Finally, you can solve the equations using
any approach that you are comfortable.
I highly recommend you learn how to use your
calculator to solve the equations using a
matrix approach.
Let’s now apply this approach to this particular
problem.
So, in this problem, you are given the angles
alpha and beta, and the mass of the crate.
Let us write this down.
Alpha is equal to thirty degree and beta is
equal to forty-five degree.
Since we want the weight of the crate and
we are given the mass, we need to convert
it so let’s convert the mass to weight by
using the equation w is equal to mg.
Now we need to identify what are the unknowns.
In this case, there are two unknown forces.
Let us denote these forces on the cable as
T sub AB and T sub BC
Now, to solve equilibrium problems, we need
to draw a free body diagram.
Drawing a good Free Body Diagram is the single
most important step in solving statics problems
so let us take a closer look at drawing free
body diagram first.
I am going to use a step by step process to
draw the free body diagram.
Step by step approach will make it easier
for you to solve any problem.
In order to draw the free body diagram, isolate
the body or area of interest.
The crate is in equilibrium, and if you look
at the crate closely you will realize there
are three cables and all we need to do is
to draw the free body diagram containing these
three cables.
And these three cables are intersecting at
point B. Let us identify this area.
Now draw an outline of the isolated part.
This outline has three lines coming out of
point B. Each line represents the ropes.
Now let’s show all the forces, both known
an unknown, on the diagram and label them.
You know force is a vector and vector is shown
using an arrow.
In this case, the arrow will be going outward
because the cables are always in tension.
So, we have now arrows representing the forces
experienced by the three ropes.
It is not enough to just show the arrows.
We must give appropriate labels for the forces.
In this case, I am going to write T sub AB
for tension in cable AB.
T sub BC for tension in cable BC, and I am
going to write W for the weight.
Weight is known and the two tensions are unknown.
So here we have a diagram that shows all the
forces.
The next step is to show all the necessary
dimensions.
We need to write them down.
The critical dimensions in this case are the
two angles, alpha and beta.
Alpha is equal to thirty degree and beta is
equal to forty-five degree.
Weight is 100 kg. and we can convert that
to force by multiplying by g, which is 9.81,
and therefore the weight is 981 Newton.
Finally, we always need an appropriate coordinate
system to solve equilibrium problems.
In this case let us attach a rectangular XY
coordinate system with its origin at point
B.
Please make sure you always draw a separate
Free Body Diagram.
Do not ever draw a diagram right on top of
the picture that is given to you.
Now that your free body diagram is complete,
we can write the equations and then solve.
Let’s write the equation.
There is one equation.
Sum of all the forces is equal to zero.
This is a vector equation.
We can rewrite this vector equation as two
scalar equations for each axis.
Let us sum up all the forces along x axis
first.
TAC times cos 30 degree – TAB times cos
45 degree is equal to zero.
This is our first equation.
Notice the negative sign for the force T sub
AB because the tension is in the other direction.
Let us now sum up the forces along y axis.
TAC times sin 30 degree + TAB times sin 45
degree minus 981 is equal to zero.
This is our second equation.
So, we have two equations and we have two
unknowns so we can now solve them using any
approach.
Let’s solve this using our calculator and
write down the answer.
