Thursday, October 13, 2011

Conservation of Energy Lab

I saw a question today in twitter:

Modelers: how do you develop 1/2mv^2 from lab? how do you develop mgh from lab?
At the modeling workshop this summer, we did exactly that,  however, instead of just rehashing that post, (you can read it here), I figure I would tell you how I tweaked the experiment for my AP class. 

Since my AP-B class is a second year class, my students already have a working idea of the relationships (as time goes in and I fully switch to modeling, they should know the models) from the first year.  So instead of using the labs as a discovery of the relationships, I like to have some challenge in the lab in which the students have to predict something using their data. 

Here's the setup:
Equipment
vernier cart
vernier track
vernier spring launcher
motion detector attached to track opposite the launcher

Set up the track at an angle (ie - place a book under one end of the track)

Using LoggerPro and the motion detector, pull the cart back to compress the spring and let go.  Stop the detector after the cart has reached it's highest point on the track. 

The Analysis:
Have LoggerPro display a position vs time and a velocity vs time graph.  From the velocity vs time graph, highlight the data, and use the "Statistics" function.  The minimum value will be the compression ($\Delta x$), and the max value will be the maximum displacement ($d_{max}$).  Highlight the data from the velocity vs time graph, and the maximum value is the maximum velocity ($v_{max}$). 

(Note- if you want to do so, you can have the kids look at what position the max speed occurs (x=0)

Repeat the procedure several times, recording $\Delta x$, $v_{max}$, and $d_{max}$ into a second data set.  Plot $d_{max}$ vs $\Delta x$.  Have the students linearize this first graph, and they should see that $d_{max}$ is proportional to $\left( \Delta x \right)^2$.  Now plot $v_{max}$ vs $\Delta x$, lead student to plot $\left( \Delta x \right)^2$ on the x axis, since that will allow this graph to relate to $d_{max}$.  They should find that they need to plot $v_{max}^2$ on the y axis.

(If you want to take it a step further and include the masses to fully develop conservation of energy, go for it.  As I said, my kids already knew those relationships from last year)

So here was my twist, how do you relate the maximum displacement to the vertical height?  Since my students knew the energy relationships, I had them use their data and trigonometry to calculate the angle of the track.  Just to give you heads up, here is what they should get...

From trigonometry, you know:
$h_{max}=d_{max}sin \theta$

And since mass is used for both kinetic and gravitation energy, you can rewrite the energy conservation as:

$g*d_{max}sin \theta = \frac{1}{2}v_{max}^2$

Therefore:

$\large \theta=arcsine \left(\frac{g*d_{max}*v_{max}^2}{2}\right)$

I then measured the angle of the track using a level app in my iPhone to compare the actual angle to the one predicted by the groups.  The app I has was able to measure to the tenth of a degree.  Most groups were able to get within $0.5^o$ of the value measured on my iPhone.

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