- Good morning.
Today, we are going to review simple harmonic motion
as part of the AP Physics C mechanics curriculum.
♫ Flipping Physics ♫
An object is in simple harmonic motion if the acceleration
of the object is proportional to the displacement
of the object from equilibrium position,
and if the acceleration of the object is directed
toward the equilibrium position.
For example.
A horizontal mass-spring system on a frictionless surface
has a free-body diagram which looks like this.
The force normal is up, the force of gravity is down,
and the force of the spring is directed to the left,
toward the vertical dotted line, which is called
the equilibrium position, or the rest position,
where the end of the spring would be
if the block were at rest.
Bobby, please sum the forces in the x direction
on the block and solve for the acceleration of the block.
- Sure.
The net force in the x direction equals the force
of the spring, which is negative, because it is to the left,
which equals mass times acceleration in the x direction.
The magnitude of the equation for the force of a spring
is the spring constant times the displacement
from equilibrium position, or K times x.
Divide by mass, and we have the acceleration
in the x direction equals the negative of
the spring constant, divided by the mass,
all multiplied by x.
- Great.
Notice how the acceleration is proportional to the
displacement from equilibrium position.
In other words, as the displacement increases,
the magnitude of the acceleration also increases,
and because of this negative, the direction of
the acceleration is opposite of the direction
of the displacement from equilibrium position.
In other words, the acceleration is directed
toward the equilibrium position.
Which means this equation fits the conditions
for simple harmonic motion, which is why this block
will continue to slide back and forth indefinitely
in the absence of friction.
Because amplitude is defined as the maximum displacement
from the equilibrium position, the maximum acceleration
of a mass-spring system is equal to the spring constant
divided by the mass, all multiplied by the amplitude.
Billy, what is the equation for instantaneous acceleration
in terms of position?
- Uh, well, we know instantaneous acceleration
equals the derivative of velocity as a function of time,
and instantaneous velocity equals the derivative
of position as a function of time.
That means acceleration equals the derivative with respect
to time of the derivative of position with respect to time,
or what we call the second derivative of position
with respect to time.
- Then we can replace the acceleration in the previous
equation for a mass-spring system with the second derivative
of position with respect to time.
And we let the spring constant divided by the mass
be equal to omega squared, where omega is defined
as the angular frequency.
We get the second derivative of position with respect
to time is equal to the negative of the square
of the angular frequency times position.
This is the condition for simple harmonic motion.
And this equation is not on the AP equation sheet,
you need to memorize it.
For a mass-spring system, the angular frequency
is equal to the square root of the spring constant,
divided by the mass, and omega is also equal to
the change in angular position over change in time,
and for one full cycle, the change in angular position
is two pi radians, and the change in time is the period,
which is capital T.
(sneezes)
- Bless you, Mr. P!
- [Mr. P] Thank you.
Solving for the period gives us the period
of a mass-spring system, which does appear
on the AP equation sheet, and I definitely recommend
you know and understand this derivation.
In addition, you should know and understand the derivation
of the period of a pendulum, which is equal to two pi
times the square root of the length of the pendulum,
divided by the acceleration due to gravity.
- [Bobby] Mr. P? - Yes, Bobby?
- We should clarify that the length of the pendulum
is from the center of suspension to the center of mass
of the pendulum, right?
- Yes, Bobby, we certainly should.
- [Bo] Mr. P? - Yes, Bo?
- Are frequency and angular frequency the same thing?
- Bo, no, frequency and angular frequency
are not the same thing.
Frequency, or lowercase F, is the number of cycles
an object goes through per second, and the frequency
is equal to the inverse of the period.
And we've already shown that the angular frequency
equals two pi divided by the period, therefore
angular frequency equals two pi times the frequency.
So, again, no, frequency and angular frequency
are not the same thing, however clearly they are related.
An equation for position as a function of time,
which appears on your AP equation sheet, which satisfies
the condition for simple harmonic motion is,
the position as a function of time equals amplitude
times the cosine of the quantity, angular frequency
times time plus phi.
Because angular frequency is in radians per second,
you must use radians for phi in this equation.
Speaking of phi.
Phi is the phase constant, or, the phase shift.
For example, if you look at the graph of Y equals
cosine theta, with theta in radians, and compare that
to the graph of Y equals the cosine of the quantity
theta, plus a phase constant of pi over two radians,
you can see that the cosine graph is phase shifted
to the left by pi over two radians,
which is also 90 degrees.
That is what the phase constant does.
It shifts the location of the cosine or sine wave
on the horizontal axis.
Billy, please take the derivative of our position
as a function of time equation for simple harmonic motion.
- Well, velocity equals the derivative of position
as a function of time.
substituting in the equation for position gives us
the derivative with respect to time of amplitude times
cosine of angular frequency times time
plus the phase constant.
Taking the derivative, we need to use the chain rule.
Amplitude is constant, so just leave that out front.
The derivative of cosine is negative sine, so we have
the negative sine of angular frequency times time
plus the phase constant.
And then we need to multiply by the derivative
with respect to time of the angular frequency times time
plus the phase constant, which is just angular frequency.
So, the simple harmonic motion equation for velocity
with respect to time equals the negative of the amplitude
times angular frequency times the sine of the angular
frequency times time plus the phase constant.
- Correct.
And because sine has a maximum value of one and a minimum
value of negative one, the maximum velocity of an object
in simple harmonic motion equals the amplitude
times the angular frequency.
Bo, could you now please determine the equation for
the acceleration of an object in simple harmonic motion?
- Acceleration equals the derivative of velocity
with respect to time, so we have the derivative
of the velocity equation we just determined.
Amplitude and angular frequency are constants,
the derivative of sine is cosine, and again, we multiply
by the derivative of omega t plus phi, which is just omega.
So, the acceleration as a function of time for an object
in simple harmonic motion is negative amplitude
times angular frequency squared, times the cosine
of angular frequency times time plus the phase constant.
- Great.
And again, because cosine varies between one
and negative one, the maximum value for the acceleration
of an object in simple harmonic motion is equal to
the amplitude times the angular frequency squared.
And now, let's bring this all the way back to the condition
for simple harmonic motion.
If we rearrange the equation for the acceleration of
an object in simple harmonic motion just a little bit,
we can show that the acceleration is equal to the negative
of the square of the angular frequency
times the equation for position.
Which means the original equation for position
which was given on the AP equation sheet
fits the condition for simple harmonic motion.
Now, let's take a look at the graphs of these equations.
Position is a cosine graph, velocity is a negative
sine graph, and acceleration is a negative cosine graph.
And, let's return back to the horizontal mass-spring system
and relate it to these graphs.
If the block is at its maximum displacement from
the equilibrium position to the right,
this distance is called what, class?
- [All] Amplitude.
- Let's call this position number one.
The block will then move to the left and eventually be
at the equilibrium position, and let's call this
position number two.
The block will pass through position number two
and eventually reach its maximum displacement
from equilibrium position, or amplitude, on the left.
Let's call this position number three.
The block will then slide back to position two,
and then back to one, then two three one two
three two one two, and in the absence of friction,
this motion will never stop.
This block is in simple harmonic motion.
Bobby.
At position two, what do we know about the velocity
of the block?
- Well, according to the velocity graph, at position two,
the block will either have its minimum or its maximum value.
- In other words, the magnitude of the velocity
will be a maximum.
- And we proved earlier, it will have a value of
the amplitude times the angular frequency.
- Billy, what about the velocity of the block
at position one?
- Actually, the velocities at positions one and three
will be the same, they will both be zero.
You can see this from the graph, because the curve
passes through the time axis at positions one and three,
and we also know this because the block will be changing
directions at positions one and three, and when the velocity
changes directions, its value has to pass through zero.
- Bo, what about the acceleration at all three positions?
- Well, the acceleration at position two,
which is the equilibrium position, is zero.
You can see that from the graph, because the curve
passes through the horizontal axis at position two.
But we also know it because the force of the spring
will be zero at the equilibrium position, so the net force
in the x direction will equal zero, so the acceleration
will be zero.
The magnitude of the acceleration will be a maximum
at both positions one and three, because the displacement
from equilibrium position is the amplitude there,
so the magnitude of the net force will be a maximum,
so the magnitude of the acceleration will be a maximum.
And we determined its value just a little bit ago,
the maximum acceleration is amplitude
times angular frequency squared.
- Thank you everybody, that's great.
Now, an important question.
Is simple harmonic motion also uniformly accelerated motion?
- I don't think so? - Probably not?
- No, if you look at the graph of acceleration as a function
of time, you can clearly see the acceleration is not
a horizontal line, so it is not a constant acceleration,
so it is not uniformly accelerated motion.
- Please remember, simple harmonic motion is not
uniformly accelerated motion.
Okay, class, is mechanical energy conserved
in simple harmonic motion?
- [Bobby and Bo] Yes. - [Billy] Yes, it is!
- Yes, because no energy is added to or removed from
the system via a work done by an external force,
and because no energy is dissipated via a work done
by friction, mechanical energy is conserved
during simple harmonic motion.
Notice when the block is in positions one and three,
it is located its maximum displacement from equilibrium
position, or the amplitude, and therefore it is at rest
and has no kinetic energy, therefore the only energy it has
is elastic potential energy, therefore the total mechanical
energy equals one half times the spring constant
times the amplitude squared.
And when the block is at position number two,
or equilibrium position, the only energy it has
is kinetic energy, so the total mechanical energy
also equals one half times mass
times the maximum velocity squared.
This brings us to the end of my review of simple harmonic
motion, next you can enjoy my review of equations
to memorize for the AP Physics C mechanics exam,
or you can visit my AP Physics C review webpage.
Thank you very much for learning with me today,
I enjoyed learning with you.