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Do you want to impart the idea that the k in the static spring is
different from the k in the oscillating spring, or do you want to say
that the k belongs to the spring and something else (like an unknown
moving mass contribution) is making them appear to be different?
If you prefer the latter approach, then when you discuss the oscillating
behavior (period = 2pi\sqrt{m/k} for example) ask them what the "m"
actually represents. They should distinguish this from the masses used in
the static case if you present the static as |\Delta F |= k |\Delta X|
for a linear spring where you begin \Delta X=0 with a stretched spring.
(Be aware that springs which have no gap between the coils may actually
be stretched from their real zero force lengths. That's why I start with
a slightly stretched spring.)
When they figure out that part of the spring is oscillating, then you can
set m = m_bob + m_unknown, take several data points, plot period^2 versus
m_bob. The ratio of the T^2 intercept to the slope yields the
contribution of the spring's mass, m_unknown.
I don't think this is too hard for HS students and it gives a good
opportunity for modeling an unknown and then extracting it. It's also an
example of "linearizing" a system which is a good first order tool.
Bn
X6588
-> -----Original Message-----
-> From: Phys-l [mailto:phys-l-bounces@www.phys-l.org] On Behalf Of Joseph
-> Bellina
-> Sent: Monday, February 09, 2015 8:06 AM
-> To: Phys-L@Phys-L.org
-> Subject: Re: [Phys-L] determine k
->
-> Why not have the students measured the same k from frequency with the
-> identical setup 5 times the find the mean and the average deviation.
Use
-> that as the uncertainty Then have them find k from elongation 5 times
find a
-> mean and average deviation Then suggest that if the ranges defined by
the
-> average deviations don't overlap you could argue that the difference
is real
-> the k's are different. If they do overlap they could be the same Groups
-> using a small mass in the oscillation experiment might see a
difference.
-> Groups using a large mass might not depending in tge mass of the
spring If
-> there is a real difference you could then have a discussion of why the
to
-> methods gave different results and lead tge students to see that the
mass
-> on the end isn't the only thing that is accelerating. Having
congratulated
-> them for that insight you could tell them about the 1/3 spring mass
-> correction from theory and see if that accounts for the discrepancy
->
-> Might be worth a try
->
-> Joe
->
-> Sent from my iPhone
->
-> > On Feb 9, 2015, at 3:17 AM, John Denker <jsd@av8n.com> wrote:
-> >
-> >> On 02/08/2015 09:44 PM, I wrote:
-> >> I think the process outlined above is pretty much squeaky-clean in
-> >> terms of physics and statistics.
-> >
-> > Well, not quite. I thought about it some more.
-> >
-> > If you measure the spring using an oscillator, I predict that you
will
-> > discover the existence of something called /systematic error/.
-> >
-> > The textbooks are full of techniques for handling random errors ...
-> > but not so much systematic errors.
-> >
-> > In particular, I predict that your oscillators will be affected by
the
-> > mass /of the spring/ ... not just the mass of the bob. (The Hooke's
-> > law experiments will not be affected in the same way.)
-> >
-> > There is a longstanding (but not venerable) tradition on phys-l of
-> > making things overly complicated. I'm trying to avoid that, or at
-> > least minimize the damage.
-> >
-> > The oscillator experiment has the /potential/ to be extremely
-> > accurate. Mass can be measured very precisely, and time can be
-> > measured even better than mass. It pains me to think of a
potentially
-> > excellent experiment spoiled by systematic error.
-> >
-> > If this were a college experiment instead of a HS experiment, the
path
-> > would be straightforward:
-> > measure the oscillator frequency using several different bobs, with
-> > widely varying masses. Then do some data reduction to model the
-> > contribution from the mass of the spring.
-> >
-> > For HS that's way too much work. It's too much lab work and too much
-> > cognitive load. I'm trying to come up with some constructive
-> > suggestions, with limited success. Right now I'm thinking out loud.
-> >
-> > Of course the simplest thing is to just take the attitude that "life
-> > sucks and then you die".
-> > That is, just accept that the dominant error will be a systematic
-> > error.
-> >
-> > Alternatively, one could imagine just using a "big enough" mass so
-> > that the spring is negligible in comparison. Alas, I doubt that can
-> > be done with real-world springs ... not without a painful sacrifice
of
-> > accuracy, or a more complicated experimental apparatus, or some other
-> > nasty business.
-> >
-> > A reasonably simple way to get decent accuracy is to weigh the spring
-> > and use theory to say that the /effective mass/ is the mass of the
bob
-> > plus 1/3rd of the mass of the spring.
-> >
-> > You could just pull that result out of some place where the sun
-> > doesn't shine, or you can justify it empirically: Have various
-> > students use different masses. We can't expect the students to model
-> > the data properly, but the teacher can do it. The frequency should
-> > fall on a straight line as a function of 1/sqrt(effective mass).
-> > Here's a spreadsheet:
-> > https://www.av8n.com/physics/measure-k-oscillator.xls
-> > By wiggling the scrollbar you can show that the data fits the model
-> > much better if you include 1/3rd of the spring in the effective mass.
-> >
-> > My spreadsheet uses synthetic data, but you could perfectly well
-> > populate it with real data (in the "bob mass" and "observed freq"
-> > columns) and use it to fit to the effective mass.
-> >
-> > I know this is still too complicated, but it's the best I can do at
-> > the moment. (There's a long list of even-more-complicated schemes I
-> > considered and rejected.)
-> >
-> > There is lots of upside potential here. At the college level this is
-> > a nifty way to demonstrate why you should not put much faith in
-> > chug-and-plug statistics. You could measure the frequency to one
part
-> > in a gazillion, and you could "prove"
-> > that the statistical uncertainty was very small ... but there would
-> > still be systematic error, and statistics generally won't tell you
-> > that, especially if you use only one bob. If you use a variety of
-> > bobs, you will begin to see some scatter in the data, but even so,
the
-> > mean of the distribution will *not* be a good estimate of the thing
-> > you are trying to measure.
-> >
-> > Any decent statistician will tell you that statistics by itself is
-> > never enough. Doing statistics on a black box is a recipe for
-> > disaster. You need to do the physics. That is, you need to build a
-> > model that makes sense in fundamental physics terms. Then analyze
the
-> > data by fitting to the model.
-> >
-> > This is fairly typical:
-> > a) If you do the statistics incautiously, you get badly fooled.
-> > b) If you do the statistics more carefully (e.g.
-> > using an ensemble of bobs), statistics will tell you that you've
got
-> > a problem, but won't tell you how to fix it.
-> > c) To fix it, you need physics. (Of course you still need the
-> > statistics.)
-> >
-> > If you know enough theory, a zero-parameter model suffices for this
-> > experiment: You know that for an ideal spring the fudge factor is
-> > 1/3rd of the spring mass. Alternatively, you can skip the theory and
-> > determine the fudge factor empirically, using a one-parameter model
-> > (and modestly more data).
-> >
-> > _______________________________________________
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