This lecture is going to be about simple harmonic motion, which has a very precise
definition that's going to be important to understand. The definition is: periodic
motion around a central point of equilibrium where there is no net force.
There is a restoring force that acts in the opposite direction proportional to
the displacement from the equilibrium position, and acts in the direction
opposite to that displacement. So that definition is pretty advanced. I'm going
to take you through it one step at a time. Two of the most common objects that
occur in simple harmonic motion are the spring and pendulum, and both of
these have what's called an "equilibrium position." And at the equilibrium position
there is no net force, that's one of the definitions of simple harmonic motion.
There has to be a certain point where there's no net force on the object from
the system. So you can see there are different reasons for why this is the
case. With the spring that's just the spring's natural length, so it doesn't put
a push or a pull on the orange box. And on the pendulum at that exact point the
force of gravity is being perfectly balanced up by the force of tension, so
there's no net force at that point either, so this is the first big idea of
simple harmonic motion: any object in simple harmonic motion goes back and
forth around an equilibrium position where there's no net force, so these
objects are going to get farther away from and then closer to that equilibrium
position as they move back and forth. So as this spring is displaced farther away
from its equilibrium position, you can see that there is a bigger and bigger
spring force pulling the object back toward the equilibrium position. So the
farther away you go from the equilibrium position, the more the spring tries to
push or pull you back to that equilibrium position. In a previous video
on elastic potential energy I talked about Hooke's Law. I'm going to bring that
up again here, and I'll link that other video in the description of this video.
So when we're describing a spring, delta X is the displacement of the spring from
its equilibrium position. Fs is the force applied by the spring on the object, and
in that Hooke's law equation that you can see at the bottom. The spring force is
equal to the spring constant times x, and that's negative just
because the force always points in the opposite direction as the displacement
from the equilibrium position. In the picture in the slide you can see that
the displacement is to the right, but the force is pulling it back left. That makes
sense. However you displace the spring it's going to pull opposite to that
displacement, so looking back up at my definition of simple harmonic motion, the
definition says that there is a restoring force that acts on the object
that is linearly proportional to the displacement from the equilibrium
position and acts in a direction opposite to that displacement. So here I
can see that Fs is playing that role. It gets bigger as the displacement from the
equilibrium position gets bigger, and it pushes back toward the equilibrium
position in the opposite direction of the displacement. And Hooke's law is
linear because it obeys the form y = mx + b you can see that it
fits that form. So if you were to graph that on a linear graph, -k would
be your slope and 0 would be your y-intercept. So a spring fits the
definition of simple harmonic motion. There's this restoring force that gets
bigger in a linear way compared to the displacement. There's a linear
relationship between the force that pulls the object back and the
displacement away from the equilibrium position. So we're now going to look at
the pendulum. So the pendulum's displacement occurs at a certain angle
from the equilibrium position like this, and the farther away from the
equilibrium position it gets the more unbalanced the force of gravity on the
pendulum will be. I'm going to show you why using this animation so you can see
that there's a force of gravity going straight down on this pendulum, and if I
split this up into an x and y component along the angle of the pendulum I can
see that this component is going to be the force of gravity times sine of the
angle of the pendulum, and the other force, the other component, is pointing in
a parallel direction to the tension of the rope of the pendulum. And because the
pendulum isn't falling up or down in that particular green direction, that
means that the forces are being balanced out, so both of these must be Fg
times cosine of the angle of the pendulum. So because those are being
balanced out, the only net force on this pendulum is Fg sine theta. That's the
force that's pulling the object back toward the equilibrium position, and just
to prove that this is linear I'm gonna write down a quick proof of this linear
relationship to the displacement from the equilibrium position. I just need to
prove to you that this definition also works for pendulums where there is a
linear relationship between the restoring force and the displacement
from equilibrium, and there's that relationship and you can see that it
fits y = mx + b. One important vocab word for simple harmonic motion
will be amplitude. The amplitude of simple harmonic motion is the maximum
displacement from the point of equilibrium that the object achieves.
This may sound a little simple but it's actually very precise. So as the spring
oscillates back and forth this green line is showing me the maximum
displacement from the equilibrium position, and as it goes in the other
direction it's displaced in that same amount, but again in the opposite
direction, so the amplitude is the maximum displacement from the point of
equilibrium. So one of these green arrows, one of those lengths would be the
amplitude, because it's the line drawn from the equilibrium point to the far
edge. Some students get confused by this and assume that the entire length of the
path, the full forward and back motion, is the amplitude, but this is not the
amplitude, that's actually twice as large as the amplitude because the amplitude
is only from the equilibrium position to one end, not the entire length. All objects
in simple harmonic motion have a period and a frequency. Just as a reminder period
is the amount of time an object takes to complete one full cycle and it's
measured in seconds, whereas the frequency is the amount of repetitions a
cycle completes in one second and it's measured in Hertz. And Hertz or Hz
really just means events per second. And because of their definition period
is equal to one over frequency. So we can calculate the period and frequency of
objects in simple harmonic motion, and in the simple harmonic motion the period
is specifically the amount of time it takes the oscillation to go to both
extremes and then go back to its starting point, not just to one extreme
and back. So I'll show you what I mean by that. I'm going to start to time the
oscillation of this spring. So this is the starting point of the spring.
This is one extreme. it goes back to the starting point, gets to the other extreme
of its motion, and then comes back. So that was one full period of motion and I
can see that that took 12 seconds of oscillation, so the period would be 12
seconds. One common mistake is for students to start the object at the
starting point, allow it to oscillate to one extreme and then go back, and stop
the time as soon as it gets back to its starting point. But this would not be
correct. This says that the period is 6 seconds, which is only half of the full
period, so this is what I mean by the amount of time it takes the oscillation
to go to both extremes and then back, rather than just to one extreme and then
back to its starting point. So the period of this spring would definitely be 12
seconds, not 6 seconds. we use many different types of graphs to show many
specific properties of objects in simple harmonic motion at different
points along their path. We'll start with the position-time graph of an object in
simple harmonic motion. So what I'm going to be doing here is measuring the
displacement of the spring from its equilibrium position as time passes. So
my y-axis shows the vertical height of the spring and the x-axis shows the
amount of time that has passed as the spring's oscillating. And as the spring
oscillates you're going to notice that the graph of its displacement from its
starting point over the time that has passed forms a sine wave.
There we go. So all graphs of simple harmonic motion will work this way.
Pendulums also produce sine wave graphs, and any other object in simple harmonic
motion also produces a sine wave or cosine wave graph. We can get some useful
properties of the motion from this graph. For example the height from the center
is the amplitude, because that's the total distance from where the center
point of the object is, where its equilibrium position is, to one extreme.
So that total height of the wave from the center to the top would be the
amplitude. Just notice that I'm not measuring from the very highest point on
the wave to the very lowest point on the wave. That would not be the amplitude
here. Just like in the example of amplitude I showed earlier. We can also
find the period of the simple harmonic motion graph. If time is on the x axis
the period is going to be 1 full completion of that sine wave going all
the way up, all the way down, and back to where it started. So again a common
mistake here would be for students to look at the sine wave and say "oh the
period is 6 because that's how long it takes the wave to get back down to its
starting position" but the period is 12 seconds specifically, because it has to
go up to one extreme, back to the other extreme and then back to its starting
point. That full wave pattern is one period, not just half of the wave pattern.
Okay, this is going to be kind of a throwback, but we're going to use the
information that the displacement time graph of an object in the simple
harmonic motion forms a sine wave to find the velocity time and acceleration
time graphs of an object in simple harmonic motion, and you'll remember that
the slope of a displacement versus time graph is equal to the velocity, and the
slope of a velocity time graph is equal to the acceleration, so looking at these
slopes I can see that the slopes of the displacement time graph start positive
and go to 0, so that means that the velocity is also going to start positive
and go to zero. Next the slopes become more and more
negative, and then become less negative until they hit zero again. So the same
thing will happen to the velocity. It becomes more and more negative and then
less negative until it hits zero again. Finally the slopes are getting bigger
and bigger and bigger, so that's what the velocity
will do until it hits some maximum. So that is what the velocity time graph of
an object in the simple harmonic motion would look like. So in this particular
case that's actually a cosine wave. We can now translate from a velocity time
to an acceleration time graph using the fact that the slope of a velocity graph
is equal to the acceleration. So looking at the slopes here I can see that the
slope starts at zero, becomes very negative, goes back to zero, becomes very
positive, and hits zero again. So this is what the acceleration graph will look
like. So now we can use these graphs to find more information about objects in
simple harmonic motion. One thing I notice is that the velocity is highest
at the equilibrium point, and zero at the extrema, so the equilibrium point is
whenever the displacement from the equilibrium point is equal to zero. So
there are three places on these graphs where the object hits the equilibrium
point. Here, here, and here, and you can see that when this happens, when the
displacement is zero, the velocity is actually at a maximum size. It's either
very positive or very negative, but it's never zero, and here at the extrema where
the displacement is at its largest. I can see that the velocity hits zero, so this
fits our experience of simple harmonic motion because when this object is
moving in simple harmonic motion it always moves very fast through the
equilibrium position. So that's where the v-max will occur, but then as it gets
closer to that maximum or minimum its velocity goes to zero, and then it goes
very fast and then it hits zero at the other maximum again, and it keeps
oscillating up and down like that. So the fastest velocity always occurs at the
equilibrium position and the velocities of zero occur at the extrema of the
motion. We can also see that the acceleration is highest at the extrema
where the object is not moving and zero at the equilibrium point where the
object is moving the fastest. I found that this one is a little more difficult
for students to understand, and it's understandably difficult because it's
moving very fast through that equilibrium point ,but what this is
saying is that the acceleration is zero where the velocity
is at its highest value. So whenever the velocity hits a maximum at the
equilibrium position the acceleration actually hits zero, so I'll show you why
that is in just a moment. What's extra confusing is the places
where the velocity is zero are actually the places where the acceleration is the
highest. Again I'll show you why that is in just a moment.
So looking at this spring I know that here the acceleration is zero, and up
here the acceleration is at its max pulling it back down, and over here at
the other extrema the acceleration is at its max pushing it back up.
So if we think about where the forces are occurring on this object in simple
harmonic motion it makes sense that at the equilibrium position there's no
force. So as a result there's no acceleration either, and at the maximum
points the force is going to be at a maximum pushing the object back toward
the center, so that means the acceleration will also be a maximum
pushing the object back toward the center. Another way to understand this is
that acceleration being zero absolutely does not mean that the object is not
moving. Acceleration is a change in velocity so if the acceleration is zero
that just means that the velocity is not changing as it coasts very quickly toward
the equilibrium position it's just going through unrestricted, so it just moves at
a high constant velocity until it gets to the other side of the equilibrium
position and begins to slow down due to the acceleration pointing in the
opposite direction, and the point where the velocity is changing the most is
where it hits zero, changing from positive to negative, so that's where the
acceleration is at a maximum. So that's where the acceleration is at a maximum,
at the two maximums of the movement of the displacement, and the acceleration
points in the opposite direction of the displacement. So those are all really
important central facts to know about simple harmonic motion going forward.
Just have a few more basic ideas to cover. First of all, if we made a graph of
the force on this object in simple harmonic motion it would look the same
as the acceleration graph, just with an M multiplied by all the values because
force is directly proportional to acceleration. Energy graphs are also
interesting and important in simple harmonic motion. Energy is a scalar and
cannot be negative, kinetic energy is proportional to velocity, and potential
energy is proportional to displacement, so that means that if we were to graph
the kinetic energy and potential energy of this spring, that same spring we were
graphing before it would look something like this and like this. So the farther
away you are in equilibrium the more potential energy you have, and the faster
you're moving the more kinetic energy you have, and these graphs can't go below
zero. They can just be either very positive or zero. One interesting pattern
here is that because of the conservation of energy, if you add up the kinetic
energy and the potential energy together you'll notice that their total energy
always stays the same. If you add the height of the red line and the green
line at any point you'll notice that it always makes the exact height of that
purple line which is just a flat constant line. The reason why this is
happening is because energy cannot be created or destroyed. There's no outside
work being done on this object in simple harmonic motion, so that means that all
of the energy is staying inside of the system. So whenever it gains kinetic
energy, whenever it goes faster, it loses potential energy. It becomes
closer to the equilibrium position. For springs the potential energy we're
talking about is elastic potential energ,y and for pendulums the potential
energy is gravitational potential energy. So which type of potential energy we use
will be different depending on the object, but the kinetic energy is always
just kinetic energy. And that's everything that you have to know about
basic simple harmonic motion for this course.