Welcome back.
And if you were covering your eyes because you didn't want
to see calculus, I think you can open your eyes again.
There shouldn't be any significant displays of
calculus in this video.
But just to review what we went over, we just said, OK if
we have a spring-- and I drew it vertically this time-- but
pretend like there's no gravity, or maybe pretend like
we're viewing-- we're looking at the top of a table, because
we don't want to look at the effect of
a spring and gravity.
We just want to look at a spring by itself.
So this could be in deep space, or something else.
But we're not thinking about gravity.
But I drew it vertically just so that we can get more
intuition for this curve.
Well, we started off saying is if I have a spring and 0-- x
equals 0 is kind of the natural resting point of the
spring, if I just let this mass-- if I didn't pull on the
spring at all.
But I have a mass attached to the spring, and if I were to
stretch the spring to point A, we said, well what happens?
Well, it starts with very little velocity, but there's a
restorative force, that's going to be pulling it back
towards this position.
So that force will accelerate the mass, accelerate the mass,
accelerate the mass, until it gets right here.
And then it'll have a lot of velocity here, but then it'll
start decelerating.
And then it'll decelerate, decelerate, decelerate.
Its velocity will stop, and it'll come back up.
And if we drew this as a function of time,
this is what happens.
It starts moving very slowly, accelerates.
At this point, at x equals 0, it has its maximum speed.
So the rate of change of velocity-- or the rate of
change of position is fastest. And we can see the slope is
very fast right here.
And then, we start slowing down again, slowing down,
until we get back to the spot of A.
And then we keep going up and down, up and down, like that.
And we showed that actually, the equation for the mass's
position as a function of time is x of t-- and we used a
little bit of differential equations to prove it.
But this equation-- not that I recommend that you memorize
anything-- but this is a pretty
useful equation to memorize.
Because you can use it to pretty much figure out
anything-- about the position, or of the mass at any given
time, or the frequency of this oscillatory motion, or
anything else.
Even the velocity, if you know a little bit of calculus, you
can figure out the velocity at anytime, of the object.
And that's pretty neat.
So what can we do now?
Well, let's try to figure out the period of
this oscillating system.
And just so you know-- I know I put the label harmonic
motion on all of these-- this is simple harmonic motion.
Simple harmonic motion is something that can be
described by a trigonometric function like this.
And it just oscillates back and forth, back and forth.
And so, what we're doing is harmonic motion.
And now, let's figure out what this period is.
Remember we said that after T seconds, it gets back to its
original position, and then after another T seconds, it
gets back to its original position.
Let's figure out with this T is.
And that's essentially its period, right?
What's the period of a function?
It's how long it takes to get back to your starting point.
Or how long it takes for the whole cycle to happen once.
So what is this T?
So let me ask you a question.
What are all the points-- that if this is a
cosine function, right?
What are all of the points at which cosine is equal to 1?
Or this function would be equal to A, right?
Because whenever cosine is equal to 1, this whole
function is equal to A.
And it's these points.
Well cosine is equal to 1 when-- so, theta-- let's say,
when is cosine of theta equal to 1?
So, at what angles is this true?
Well it's true at theta is equal to 0, right?
Cosine of 0 is 1.
Cosine of 2 pi is also 1, right?
We could just keep going around that unit circle.
You should watch the unit circle video if this makes no
sense to you.
Or the graphing trig functions.
It's also true at 4 pi.
Really, any multiple of 2 pi, this is true.
Right?
Cosine of that angle is equal to 1.
So the same thing is true.
This function, x of t, is equal to A at what points?
x of t is equal to A whenever this expression-- within the
cosines-- whenever this expression is equal to 0, 2
pi, 4 pi, et cetera.
And this first time that it cycles, right, from 0 to 2
pi-- from 0 to T, that'll be at 2 pi, right?
So this whole expression will equal A, when k-- and that's
these points, right?
That's when this function is equal to A.
It'll happen again over here someplace.
When this little internal expression is equal to 2 pi,
or really any multiple of 2 pi.
So we could say, so x of t is equal to A when the square
root of k over m times t, is equal to 2 pi.
Or another way of thinking about it, is let's multiply
both sides of this equation times the inverse of the
square root of k over m.
And you get, t is equal to 2 pi times the square root-- and
it's going to be the inverse of this, right?
Of m over k.
And there we have the period of this function.
This is going to be equal to 2 pi times the square
root of m over k.
So if someone tells you, well I have a spring that I'm going
to pull from some-- I'm going to stretch it, or compress it
a little bit, then I let go-- what is the period?
How long does it take for the spring to go back to its
original position?
It'll keep doing that, as we have no friction, or no
gravity, or any air resistance, or
anything like that.
Air resistance really is just a form of friction.
You could immediately-- if you memorize this formula,
although you should know where it comes from-- you could
immediately say, well I know how long the period is.
It's 2 pi times m over k.
That's how long it's going to take the spring to get back--
to complete one cycle.
And then what about the frequency?
If you wanted to know cycles per second, well that's just
the inverse of the period, right?
So if I wanted to know the frequency, that equals 1 over
the period, right?
Period is given in seconds per cycle.
So frequency is cycles per second, and this
is seconds per cycle.
So frequency is just going to be 1 over this.
Which is 1 over 2 pi times the square root of k over m.
That's the frequency.
But I have always had trouble memorizing this, and this.
You always [UNINTELLIGIBLE]
k over m, and m over k, and all of that.
All you have to really memorize is this.
And even that, you might even have an intuition
as to why it's true.
You can even go to the differential equations if you
want to reprove it to yourself.
Because if you have this, you really can answer any question
about the position of the mass, at any time.
The velocity of the mass, at any time, just by taking the
derivative.
Or the period, or the frequency of the function.
As long as you know how to take the period and frequency
of trig functions.
You can watch my videos, and watch my trig videos, to get a
refresher on that.
One thing that's pretty interesting about this, is
notice that the frequency and the period, right?
This is the period of the function, that's how long it
takes do one cycle.
This is how many cycles it does in one second-- both of
them are independent of A.
So it doesn't matter, I could stretch it only a little bit,
like there, and it'll take the same amount of time to go
back, and come back like that, as it would if I
stretch it a lot.
It would just do that.
If I stretched it just a little bit, the function would
look like this.
Make sure I do this right.
I'm not doing that right.
Edit, undo.
If I just do it a little bit, the amplitude is going to be
less, but the function is going to essentially do the
same thing.
It's just going to do that.
So it's going to take the same amount of time to complete the
cycle, it'll just have a lower amplitude.
So that's interesting to me, that how much I stretch it,
it's not going to make it take longer or less time to
complete one cycle.
That's interesting.
And so if I just told you, that I actually start having
objects compressed, right?
So in that case, let's say my A is minus 3.
I have a spring constant of-- let's say k is,
I don't know, 10.
And I have a mass of 2 kilograms. Then I could
immediately tell you what the equation of the position as a
function of time at any point is.
It's going to be x of t will equal-- I'm running out of
space-- so x of t would equal-- this is just basic
subsitution-- minus 3 cosine of 10 divided by 2, right? k
over m, is 5.
So square root of 5t.
I know that's hard to read, but you get the point.
I just substituted that.
But the important thing to know is this-- this is, I
think, the most important thing-- and then if given a
trig function, you have trouble remembering how to
figure out the period or frequency-- although I always
just think about, when does this expression equal 1?
And then you can figure out-- when does it equal 1, or when
does it equal 0-- and you can figure out its period.
If you don't have it,
you can memorize this formula for period, and this formula
for frequency, but I think that might be a waste of your
brain space.
Anyway, I'll see you in the next video.