mr.p: Good morning.
Today we are going to review the simple harmonic motion
topics covered on the AP Physics 1 exam.
Bo: Hey guys!
Billy: Hey Bo.
Bobby: Hi Bo.
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mr.p: A typical example of an object
in simple harmonic motion
is a horizontal mass-spring system
on a frictionless surface.
This is a mass attached to a spring.
Currently it is in the equilibrium, or rest position.
And if I take that mass and pull it to the right,
we can call this position number one.
And notice the force of the spring is to the left.
And if we let go of the mass,
it will slide to the left
and go through the equilibrium position.
Let's call this position number two.
And, at this point, the displacement
from equilibrium position is equal to zero.
Therefore the force of the spring
which is equal to the spring constant
times the displacement from equilibrium position.
Therefore, at position number two,
because the displacement from equilibrium position
is equal to zero, the force of the spring is equal to zero.
Notice this is different than our first drawing
where the velocity of the block was equal to zero.
Here the velocity of the block is actually to the left
and the block will continue to move to the left.
Let's call this position number three.
Notice, here, that the force of the spring is to the right.
Notice that the force of the spring
is always pointed towards the equilibrium position
and increases proportionally with the displacement
from equilibrium position.
This force of the spring is called a restoring force.
A restoring force is what causes simple harmonic motion.
And simple harmonic motion is the idea
that this object will go
from position one to two to three to two to one
to two to three to two to one
to two to three to two and so on
and this will continue, without friction, forever.
Also notice that the mass will never get farther
from the equilibrium position than it does
at positions one and three.
Therefore we call this maximum displacement
from equilibrium position the amplitude.
According to Newton's Second Law,
if the force of the spring is equal to zero
at position number two,
then the net force is equal to zero,
then the acceleration at position number two
is equal to zero.
We can also look at the accelerations
at position one and three.
Because the displacement from equilibrium position
is at its maximum value,
the force of the spring is gonna be at its maximum value.
Therefore the magnitude of the maximum force
will be a maximum.
Therefore, the magnitude of the accelerations
at positions one and three will be a maximum.
And the block changes directions at positions one and three.
Therefore, the velocity there will be zero,
which means the block will have a maximum magnitude velocity
at position two.
We can also look at simple harmonic motion
in terms of circular motion.
An object moving in a circle can look like this.
We can solve for the x-position of the object
moving in a circle using the cosine of theta.
By definition, the cosine of theta
equals the adjacent divided by the hypotenuse
and, therefore, it's equal to the x-position
divided by the radius.
And we get that the x-position is equal
to the radius times the cosine of the angle.
And we can use the equation for period
in terms of frequency and angular velocity
that's on your equation sheet.
Multiply this equation by angular velocity and frequency,
and we get the angular velocity
equals two pi times the frequency.
Now, this omega is actually
called the angular frequency
but you can see it comes from the angular velocity.
In fact we can use the equation for angular velocity.
Angular velocity equals angular displacement
over change in time.
Or, angular position final minus angular position initial
all divided by time final minus time initial.
And if we set our initial conditions equal to zero,
we get angular velocity
equals angular position divided by time
or multiplying by time,
that angular position equals the angular velocity
or angular frequency times time.
And combining all these equations gives us
the x-position of the particle equals
the radius times the cosine of the quantity
two pi times frequency times time.
And, if we set the radius of the circle equal
to the amplitude in simple harmonic motion, we get
the equation that is on your equation sheet
for the x-position of an object in simple harmonic motion.
Okay, let's now look at this in terms of a graph.
You can see the position graph follows exactly
what we talked about before.
The mass-spring system oscillates back and forth
between positions one, two, three, two,
one, two, three, and so on.
Yes, Billy?
Billy: The horizontal axis isn't labeled.
What is on the horizontal axis?
mr.p: It's a good question.
This is a graph of cosine of theta.
However, we showed that theta is equal
to two pi times the frequency times time.
And it is important to realize that we define the theta
in terms of radians.
Therefore, we need to use radians
in our equation for position.
And, therefore, these are not 90, 180,
270, and 360 degrees.
These are pi over two, pi, three pi over two,
and two pi radians.
Please be careful of that.
Just to be absolutely clear.
The mass starts out at position number one,
moves to position number two,
then three, then two,
and then back to position number one.
At positions one and three,
the displacement from equilibrium position of the mass
has its maximum magnitude which is the amplitude.
Therefore, the force caused by the spring
has its maximum magnitude and, therefore,
the acceleration has its maximum magnitude.
Also at positions one and three, the mass changes directions
so its velocity is zero.
At position number two, the displacement
from equilibrium position is zero.
Therefore the force of the spring is zero.
And, therefore, the acceleration of the mass is zero.
Also at position number two, the mass has its
maximum magnitude velocity.
We show that the position of the mass-spring system
is described by the equation, position equals
amplitude times the cosine of two pi
times the frequency times time.
You can see the maximum magnitude displacements
at positions one and three.
And at position number two, the mass is
at its equilibrium position.
Now, let's look at a graph of velocity.
Notice how this matches what we determined before.
The velocity is at zero at the two most extreme
of positions, positions one and three.
And it has its maximum magnitude
at the equilibrium position, or position number two.
[Bo] Oh right, and you can see if the mass
is moving to the left or to the right
based on whether the velocity is positive or negative.
[Mr. P] Bo, that's correct.
Now let's look at the graph of the acceleration of the mass.
Again, it matches what we said before.
The maximum at magnitude acceleration
is at the most extreme positions, positions one and three.
And the acceleration is zero
at the equilibrium position, position number two.
It is very important to notice
that the acceleration during simple harmonic motion
is not uniformly accelerated motion.
So you can not use
the uniformly accelerated motion equations.
It is also good to look at the mechanical energies as well.
Here is the kinetic energy of the mass.
Bobby, what do you notice about the kinetic energy
of the mass?
[Bobby] The kinetic energy is never negative
because the velocity is squared
and kinetic energy is zero
where the velocity is zero
at the two endpoints, one and three,
and a maximum where the velocity is a maximum
at the equilibrium position, or position two.
[Mr. P] Very nice Bobby.
This is the elastic potential energy of the mass.
Billy, what do you see?
[Billy] Just like kinetic energy,
the elastic potential energy is never negative
because the displacement from equilibrium position
is squared and at the equilibrium position
there is no elastic potential energy
because the displacement from equilibrium position is zero
and when the mass is at the amplitude positions,
one and three, the elastic potential energy
is at its maximum.
[Mr. P] Absolutely, Billy.
Now, notice what happens when we add kinetic energy
to the graph, in red,
and then we add the kinetic energy
and elastic potential energies together
to get the total mechanical energy
which is now on the graph in green.
Who can tell me what major Physics concept
that the total mechanical energy on this graph illustrates?
[Bo] Oh, I get it.
The total mechanical energy is constant.
[Bobby] So mechanical energy is conserved.
[Billy] Yes, there is no friction
and there is no force applied
adding energy to the system via work
so this is Conservation of Mechanical Energy.
[Mr. P] That's right, simple harmonic motion
without friction is Conservation of Mechanical Energy.
Bobby, what is the period of a mass-spring system?
Bobby: The period of mass-spring system equals
two pi times the square root of the mass of the object
divided by the spring constant.
mr.p: And Billy, what is the equation for the period
of a pendulum in simple harmonic motion?
Billy: The period of a pendulum equals
two pi times the square root of the length of the pendulum
divided by the acceleration due to gravity.
And the length of the pendulum is specifically
the distance from the center of suspension
at the top of the string to the center of mass
of the pendulum.
mr.p: And Bobby, what is the definition of the period?
Bobby: The period is the time it takes
for an object to go through one full cycle.
mr.p: Then Bo, let's say we have a pendulum
and a mass-spring system
and we take them both to the moon?
What happens to their periods?
Bo: The period of the mass-spring system will not change
because the acceleration due to gravity
is not in the equation.
However, the period of the pendulum will increase
because the period of a pendulum
is inversely-related to the square root
of the acceleration due to gravity
which is less on the moon.
mr.p: And Bobby, if instead of taking them to the moon,
we increase their masses,
what happens to their periods?
Bobby: The period of the pendulum will stay the same
because it is independent of mass.
And the period of the mass-spring system
will increase because it is directly proportional
to the square root of the mass of the mass-spring system.
mr.p: And notice that amplitude is not
in either one of these two equations.
Therefore the period of a pendulum
and a mass-spring system are independent
of their amplitudes.
We have reached the end of my review
of simple harmonic motion for AP Physics 1.
My next review lesson is about waves.
You are more than welcome to enjoy that video.
You can also go to my website,
flippingphysics.com, where you will find
all of my review lessons for AP Physics 1 organized
with lecture notes.
Thank you very much for learning with me today.
I enjoyed learning with you.