Good morning. We have already learned about simple harmonic motion and mass-spring systems; today we
are going to look at the simple harmonic motion of a mass-spring system from a calculus perspective.
Calculus! ♪ Flipping Physics ♪
Yep. We have already learned that an object
is in simple harmonic motion when the acceleration of the object is proportional to its displacement
from equilibrium and is directed opposite the object’s displacement from equilibrium.
For a mass-spring system the spring force is the restoring force which causes this acceleration.
Recall that a restoring force is always directed toward equilibrium and the spring force is always
directed toward equilibrium, which is why the spring force is the restoring force.
The net force in the x-direction then equals the spring force
and it also equals mass times acceleration in the x-direction.
We can substitute the negative of the spring constant times displacement from equilibrium in
for the spring force. The reason this is negative is because the spring force is a restoring
force and is always directed opposite from the direction of the displacement from equilibrium.
Notice the negative means the acceleration and displacement are always in opposite directions,
regardless of whether the mass is to the left or to the right of equilibrium.
Bobby, what is the maximum acceleration of a mass in a mass-spring system equal to?
If that is the equation for the acceleration of the mass
oh, the maximum distance from equilibrium is amplitude, so the maximum acceleration equals the
spring constant over mass all times amplitude. Thank you, Bobby. Bo, what are the derivative
definitions of acceleration and velocity? Acceleration equals the derivative of velocity
with respect to time and velocity equals the derivative of position with respect to time.
Correct. That means that acceleration equals the second derivative of position with respect
to time. And we can now substitute that into our previous equation. Now, this equation gives us
the general idea of the mathematical definition of simple harmonic motion, which is that the second
derivative of position with respect to time equals the negative of omega squared times position,
where omega is called angular frequency. This equation describes simple harmonic motion.
Notice the acceleration is proportional to displacement from equilibrium
and the acceleration is directed opposite to the direction of the displacement from equilibrium
which are the conditions for simple harmonic motion. For a mass-spring system,
you can see that the square of the angular frequency equals spring constant over mass.
(Mr.p?) Yes Billy?
Why is the square of angular frequency in the equation and not just angular frequency?
That is a fair question Billy. I will show why that is in just a bit, we are not quite
there yet. (oh kay)
Yeah. A position equation which satisfies the mathematical definition of simple harmonic motion
is position as a function of time equals amplitude times the cosine of the quantity angular frequency
times time plus phi, the phase constant. Billy, remind me of what phi, the phase constant is.
Absolutely! Phi is the phase constant or the phase shift of the wave. For a simple example,
we can 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 2 radians, you can see the cosine graph is phase shifted to the left by pi over 2 radians,
which is also 90 degrees. That is what the phase constant does, it shifts the location of the
cosine or sine wave along the horizontal axis. I thought phi was the “phase angle”.
I’ve heard it is called “phase delay”. Yeah, phi has a lot of names. Most often I have
heard it called “phase constant” or “phase shift”. The most important thing to know is that it shifts
the sine or cosine wave along the horiztonal axis, however, it does not change the shape of the wave.
Bo, please determine the velocity of an object in simple harmonic motion
using the position equation. Well velocity equals the derivative of position
with respect to time, so the derivative of the position equation with respect to time. Amplitude
is constant, so that comes out of the derivative. We need to use the chain rule, so the derivative
of cosine is negative sine of the same quantity as before and then the derivative with respect
to time of that quantity or angular frequency times time plus phi. The derivative of that with
respect to time is just angular frequency. So, velocity of an object in simple harmonic motion
with respect to time equals negative amplitude times angular frequency times the sine of the
quantity angular frequency times time plus phi. (And what then is the maximum velocity equal to?)
Uh the value of sine theta has a minimum of negative 1 and a maximum of positive 1.
So, the maximum velocity equals amplitude times angular frequency.
That’s correct Bo. Thanks. Bobby, please determine acceleration as a function of time.
Sure. Acceleration equals the derivative of velocity with respect to time so the derivative
with respect to time of the equation Bo just derived. This is actually very similar to what Bo
just did. Negative amplitude times angular frequency is constant over time so take that
out from the derivative. The derivative of sine is cosine and we need to use the chain rule.
Again, the derivative of that parenthetical expression with respect to time is just angular
frequency. So, the acceleration of an object in simple harmonic motion equals the negative of
amplitude times angular frequency squared times the cosine of the quantity angular
frequency times time plus phase constant. And, because the value of cosine theta varies between
negative 1 and 1, the maximum acceleration equals amplitude times angular frequency squared.
Well done Bobby. Now, who can tell me what they see when I rearrange the equation
slightly? What is between the brackets in the acceleration equation?
That is the position equation! Correct Billy. That means we have just
shown that this position equation satisfies the conditions for simple harmonic motion.
The second derivative of position with respect to time equals the negative of the square of angular
frequency times position. (Cool. Huh. Yeah.)
And Now we can see why the equation for simple harmonic motion has the square of angular
frequency in it and not just angular frequency. Oh right, I see it! It is because, when we start
with that equation for position and take the derivative of it with respect to time twice,
we get acceleration equals the negative of the square of angular frequency times position.
The square of angular frequency is right there in the derivation
of the acceleration equation! (You are welcome!) Exactly Billy. Thanks {hear Billy} Please,
remember to use radians for phi, the phase constant, and radians per second for angular
frequency. Actually, let’s talk some more about angular frequency, starting with period. Billy,
what is the period of simple harmonic motion? In simple harmonic motion, period, capital T,
would be the time interval it takes for an object to go through one full cycle.
In other words, the values of the position equation should be equal
at times t and t plus the period. Right. After one full cycle,
the position is the same. So, the position function should give equal results after time t
and time t plus the period. That makes sense. Also, after one full cycle the phase has
increased by two pi radians. In other words, the position at time t equals the position
at time t plus the period. Substituting in the equation for position in both cases, for time
t the phase shift will be 2 pi and for time t plus the period the phase shift will be zero.
In other words, we need to shift forward by 2 pi radians at time t
in order for the position to be exactly equal at time t and time t plus the period. And …
Everybody brought amplitude times cosine to the party!
Technically we have to take the inverse cosine of the whole equation,
but sure, everybody brought cosine to the party. (Close enough for dancing!)
And when we multiply through by omega on the right-hand side, you can see we can subtract
angular frequency times time from both sides. Sure.
That leaves us with 2 pi equals angular frequency times period. Therefore, period equals two pi
divided by angular frequency or, if you prefer, angular frequency equals two pi over period.
Before we just used angular velocity equals angular displacement over change in time.
For one full cycle, that equals two pi radians over the period. Right?
(True, but the way I just did it does not use the equation for angular velocity.)
Ooooh. Right.
And that makes it more clear that omega in these simple harmonic equations is angular frequency.
Honestly, I do not have a preference of which way you use to determine the equation for angular
frequency, I just think it is useful to understand both derivations of angular frequency. Speaking
of frequency. Frequency equals the inverse of the period; therefore, angular frequency equals two pi
times frequency. Which illustrates that angular frequency and frequency are not the same.
And the units for angular frequency are radians per second, however, the units for
frequency are one over seconds or cycles per second. So, even their units are different.
Correct Billy. And now let’s return to the mass-spring system. If you recall, we derived
that the angular frequency of a mass-spring system equals the square root of the spring constant over
mass. We can now substitute that into the equation we derived for the period of simple harmonic
motion and show that the period of a mass spring system equals two pi times the square root of mass
over spring constant. We have used this equation many times before. Now we know how to derive it.
The main takeaways from today’s lesson are that the mathematical condition for simple harmonic
motion is the second derivative of position with respect to time equals the negative of angular
frequency squared times position. Period equals 2 pi over angular frequency. One position equation
which satisfies the condition for the position of an object in simple harmonic motion is position
equals amplitude times the cosine of the quantity angular frequency times time plus phase constant.
(Mr.p?) Yes Bobby?
Why do you say it like that? “One position equation which satisfies the condition
for the position of an object in simple harmonic motion is” that equation. Why “one
position equation”? Are there others that work? The position equation could use the sine function
with a different phase constant instead. The sine and cosine functions have the same shape,
they are just phase shifted from one another. Oh. That makes sense. Thanks.
You’re welcome. Agreed. Thank you, Bo. We can
take the derivative of the position equation twice to derive equations for velocity and acceleration
as functions of time. And the maximum position is amplitude, maximum velocity is amplitude times
angular frequency, and maximum acceleration is amplitude times angular frequency squared.
Those are all general equations for simple harmonic motion. All you need to do is derive
what angular frequency equals for each new physical arrangement of simple harmonic motion.
For example, next time we are going to derive the period of a simple pendulum.
Thank you very much for learning with me today, I enjoy learning with you.