Alright, what is engineering?
The way we define engineering is that it is the application of science and math
to solve complex problems of relevance to society.
So today we're going to apply engineering, but not to a complex societal problem.
We're going to apply it to a physical system and we're going to try to understand that system.
So we're going to observe what's going on and see if we can come up with a description,
a mathematical description, of our system.
So let's look first at what happens when I turn the wheels to the inside
Here, watch what happens to the bar that's sliding. See it oscillating on the top?
If I change the speed of the wheels, does it change the speed of the bar?
Now, as a second set of examples here, I'm going to
turn the wheel to the outside.
And see what happens to the bar?
Didn't oscillate that time.
Okay we're going to try to understand this using principals of science and math.
The first thing we would do is start with a sketch. So we sketch the system
and we would have the bar and the wheels.
And we would define a coordinate system
where X is gonna be the motion of the bar
So we'll say X is the location of the center of mass of the bar
and we're also going to have a coordinate system where the
zero is right here in the middle between the wheels.
So then we're also going to say that the distance between the center
from center to center of the wheels is 2L.
And we have the mass of the bar, the coefficient of friction between the wheel and the bar,
and that's the system.
Now in engineering we often isolate the part we're interested in.
We call this a "freebody diagram", this is just of the sliding bar.
But to free the bar, we need to then draw all of the forces that act on the bar.
So we have to replace the wheels with a vertical force
which we'll call the normal force
in this case over here it's N sub L.
Over here there's an N sub R for the righthand side.
And then we also have the friction forces.
F sub L and F sub R.
F sub L will act in this direction if the wheels are turning towards the inside
because then the friction is going to act this way.
And on this side, this way.
We also have gravity acting on the bar so we need to put
MG or the weight acting through the center of mass
and that's gonna be acting down.
Alright, so we have a freebody diagram of the bar
And now we're ready to apply our laws, our physics laws.
Newton's second law says that the sum of the forces
acting on the bar is equal to the mass times acceleration
of the bar in the certain direction.
So we're looking in the horizontal direction so we will sum the forces
F sub L minus F sub R is equal to M times the acceleration, which is the second derivative of x.
So let me remind you a little bit about acceleration.
So if X is a function of time, it's the position of the bar,
then the first derivative of x is the velocity.
That's dxdt, the derivative of velocity is acceleration
so that's the second derivative, and we're going to call that x double dot
or x with two dots on top.
Okay so fₗ - fᵣ = mẍ.
That's our first equation. Now the other thing that you'll notice
is that the bar did not rotate
it stayed horizontal. So that means that all of the moments
that acted on this bar were balanced out.
So if I try to hold the bar here at this point where the left wheel attaches
and I say the moments have to be equal. That means
that Nᵣ times this distance 2L must be equal to the mg times the distance L + x.
So then if I solve that for Nᵣ that means that
Nᵣ = mg times L + X divided by 2L.
And I could do the same thing over here
and I would find out that Nₗ is equal to mg times L minus X over 2L.
And I can think about whether that makes sense, if I have my bar here
you can think about whether it makes sense that the Nᵣ would be greater or less than Nₗ
So is more weight over the right wheel when x is positive?
Yes, so that makes sense.
Alright so now we have an equation for Nₗ and Nᵣ as a function of X.
And we can also remember back to physics that the friction
is related to the normal force through the coefficient Mu.
So fₗ is equal to Mu times Nₗ
And fᵣ is equal to Mu times Nᵣ.
So now using all of those equations we can now rewrite our equation of motion
Newton's second law, replacing the friction forces with normal forces
and replacing the normal forces with their relationship to X.
So we have ẍ plus Mu g over L times X is equal to zero.
That's the differential equation that we need to solve.
So we need a function where X and the second derivative of X
are of opposite signs so that they can cancel out in that equation.
If you think back to trigonometry, signs and cosigns have that property.
So let's start with X equals sin of at.
So X equals sin at, the first derivative of X is a times the cosign of at.
And the second derivative of x is then gonna be minus a squared sin at.
If we put those functions back into our equation
then only two things are possible. Either the sin of at equals zero.
Or a squared is equal to Mu g over L.
Well the sin of at cannot be equal to zero for all time.
That means that a is then equal to the square root of Mu g over L.
And we now have a function X is equal to sin of the square root of Mu G over L times time.
And that is the function that describes the motion of this bar.
What if we did it the other way and looked at this motion?
Well in that case what happens, without going through all of the math all over again
The sin of friction flips, because now the friction is applying
the forces to the bar in the opposite direction.
So instead of Fₗ minus Fᵣ we now have minus Fₗ plus Fᵣ equals mẍ.
And we go through the same substitution. So the friction forces become the normal forces times Mu
and those normal forces get replaced with their functions that have X in them.
After we do all that, our differential equation for the motion this way
is gonna look like ẍ minus Mu G over L times x is equal to zero. This time we have the minus sign.
That means the function x and its second derivative must have the same sign
and be the same function in order to cancel out.
So again thinking back to functions that you know
and what has this property? The exponential function e to the x.
or e to the at.
So in this case, lets try e to the at.
The first derivative of that would be a e to the at.
And the second derivative would be a squared e to the at.
So now we can substitute those functions back into our equation of motion
and we see that in this case again
a squared must be equal to Mu g over L.
Except this time, we have the exponential function.
So x is equal to e to the square root of Mu g over L times time.
And that's the function that describes that motion there.
So lets just recap a little, why did we do this?
Well first of all, we wanted to describe the motion of this bar.
Now we know when it will oscillate versus when it will fly off.
We also know what determines the frequency of oscillation
Mu G over L. So we know how to change it if we wanted it to go faster or slower.
We could use this to design windshield wipers, or something that oscillates back and forth.
Okay so to recap the whole engineering process that we went through
First we observed the system
and then we looked at what did we know, and what did we not know,
and what were we trying to understand?
And we created variables, and then we applied our laws of physics.
In this case we applied Newton's second law but we could apply conservation of mass, momentum, or energy.
And then once we had a set of equations, we used math.
We used algebra, trigonometry, and calculus
among other things in math in our toolbox to solve the system.
Then once we have a solution, we can use that to change the variables
to change different properties.
and if we have a much more complicated system we might want to write a computer code
that has lots of variables, and we can keep changing the computer code and simulating a system.
And that's often used in Engineering before you build something or try to do a prototype.
So those are the steps that we went through, and how we got to the understanding that we have.
If you're really interested in learning more about engineering
I have a list of references, you might want to consult some of these textbooks
and I hope you will pursue engineering careers in the future.