Mr. P.: Good morning!
So, I'm going to do something a little bit different this time.
It's just going to be me and I'm just gonna extemporaneously walk
my way through a demonstration
that I want to show you about a vertical mass spring system in simple harmonic motion,
position, velocity, and acceleration as a function of time, actually collecting that data.
I don't have a script but I do have some bullet points that I want to get through.
Let's actually start right here because this is where we have to start.
♫ Flipping Physics ♫
Mr. P.: Right, you have to start there, there's no choice. Okay, so basic idea over here,
we have a mass spring system, which I'm going to set into simple harmonic motion in a moment.
You can see I have a camera specifically for that and then the screen capture you can also see on the screen.
So let's set the mass spring system into simple harmonic motion.
So here we are.
We have 305 grams attached to this spring and below there
we have a motion sensor to collect the position as a function time of this mass spring system.
Below the mass, I've actually added a playing card, believe it or not,
the reason for that is I found that the motion sensor senses the position
of the mass much easier if I have that on there because it's just a little bit larger,
I think.
So, there we go, we have the vertical mass spring system moving in simple harmonic motion, great!
Let's just review the equations real quick:
position as a function of time equals the amplitude times the cosine of the quantity
angular frequency times time plus
phi, the phase constant.
Again it could be cosine, it could be sine. It just depends on the initial condition, really.
We derived the equation for velocity, velocity equals negative amplitude times sine
of the quantity angular frequency times time plus phi,
and then acceleration equals the negative amplitude times the angular frequency squared
times the cosine of the quantity angular frequency times time plus phi, the phase constant.
Those are the equations that we've already derived.
Now, let's collect the data for position as a function of time.
So in Capstone, the program on the computer from PASCO,
you can see on the y axis we have position,
and on the x-axis we have time.
And I'm going to click record to collect position as a function of time.
Okay, so there we have position as a function of time. That's awesome.
Okay, so, you can see - let's actually adjust the
x-axis here, the x-axis scale. So now we have something that looks like that.
Now this is our cosine wave.
And this is because it started initially at time t equals 0 at the amplitude.
If instead it had started right here, we could have used the sine wave.
This would have it starting at rest position and moving upward,
we could have just used the sine wave and the phase shift would be equal to zero,
but we have it actually starting right there, and
this has a phase shift of zero and it is the cosine wave. That is the position as a function of time, great.
So, let's also get on here - let's add another graph. Let's get a graph...
we're gonna put below this and let's put the velocity as a function of time on this graph.
Again, I want to adjust the
x-axis scale here so that it matches, and you can see right there we have two cycles.
I guess that's important to realize, we have two cycles here in our graph.
So the velocity you can see is a negative sine wave. You can see it doesn't quite
approximate a sine as well as the position did I'll talk about why that is in just a moment, and we can
also then change this to acceleration as a function of time.
So now adjust the x-axis scale so it is the same.
So here you can see this is a negative cosine wave, or at least it
approximates a negative cosine wave, you can see it starts negative down here
and then goes up and comes back down.
It is a negative cosine wave, but you can see there's a lot
more error in the acceleration than there was in the position,
there was a little bit more in velocity.
And the reason for that is
all the motion sensor collects is position.
That's the only data it's able to collect.
So this program calculates the velocity from the position.
Remember velocity equals change in position over change in time,
position final minus position initial divided by change in time.
In other words, every velocity has two positions in it.
Every velocity data point has the error of two
position data points, so it has twice the error and
acceleration equals change in velocity over change in time,
so velocity final minus velocity initial divided by change in time,
So every acceleration has two velocities and hence four positions in it.
So acceleration actually has four times the error as
position, and you can actually see that error in the acceleration graph right here.
Great, alright, let's actually get rid of this acceleration graph and just return to position as a function of time.
There we have position as a function of time. Okay, so, we can now determine
the period of this motion, we can select...
click here...
We can select a certain range of data right here,
so from there to there is two full cycles, great.
Now we can bring up the data table and
we can put position and
time on our data table, and let's adjust the position and adjust the location here,
because we don't need that.
Okay, so now you can see it actually
selects the data from the graph
also in the data table, so you can see it's highlighted there.
So you can see it starts at time 0 and ends at time 2.720.
So you can see that
the change in time for two full cycles is 2.72 seconds.
And so if we take and divide that by two we get
1.36, which is the time for one full cycle, because I selected two cycles in this graph.
One other thing I want to talk about about this data right here
that I have highlighted is notice the initial position,
that was the maximum, was .401 meters, and the final after two cycles,
the final maximum position is .400 meters.
So even after only two cycles
it has actually been damped by one millimeter.
You could see clearly it's damped, simple harmonic motion right here
because it's barely in simple harmonic motion now,
but even after two full cycles, it's already damped by one millimeter.
Now, we have the period, which means we can use the period of a mass spring system
equation to determine the spring constant of this spring.
So, that's period equals 2pi times the square root of mass divided by the spring constant,
solving for the spring constant, so square the whole equation, we get
period squared equals 4pi squared times the mass divided by the spring constant,
multiply both sides by the spring constant, divide by the period squared,
we get spring constant equals 4pi squared
times the mass divided by the period squared.
So, 4 times pi squared times the mass...
Now, it's 305 grams, which is 0.305 kilograms,
divided by the period, 1.36 squared, and the spring constant of this spring
is roughly 6.51 Newtons per meter, cool.
Oh, sorry, one other cool thing, which I really think is fun. Okay.
So back to here, back to the data, I'm going to now add a best fit sine wave.
So, the data has now - it's added this best fit line.
You can see it right there - this best fit curve, and it's a sine wave and it's of the form
amplitude times the sine of angular frequency times time plus phi,
the phase constant, plus C, and that constant C just raises it above the
time axis because all the positions are above zero.
So you can see the angular frequency is in there.
It's 4.66, and remember
angular velocity equals change in theta over change in time, if the change in theta is
one revolution, or two pi radians, the change in time is the period which is capital T, so we know
period equals 2pi divided by the angular frequency.
So, we can take 2 times pi and divide it by the angular frequency of this best fit curve of
4.66. and we will also get the period.
And the period is roughly 1.35 seconds, which is really very close to what we calculated
using two cycles, 1.36 seconds, which I think is pretty awesome.
Alright, so that's all I have for this, so I hope you enjoy because this was great.
So thank you very much for learning with me today, I enjoyed learning with you.