Lab 1: 27 Feb 2017: Deriving a power law from inertial pendulum
Lab 1: Finding the Relationship between Mass and Period for an Inertial Balance
Amy Chung and Chris Ceron
February 27, 2017
Today's Lab purpose was to find the relationship between mass and period of oscillation due to the mass' weight on the inertial balance.
Theory:
The experiment performed today was weighing an object and trying to calculate its period of oscillation. In this lab we tried to calculate a correlation from multiple known masses and their period of oscillation on an inertial balance to derive a formula that calculates an unknown mass. 
Apparatus and Procedure:
Today's apparatus was made up of a clamp, a logger pro, laptop, photogate, and an inertial balance, and 8 different weights of 100 g each. We created our apparatus by clamping the inertial balance to the table by using the clamp. Next to our table was a stand where the tip pf the balance met the photogate sensor. We placed a strip of tape to the balance so as the balance moved back and forth the sensor was able to measure the oscillations from our added masses. The sensor is plugged into our laptop and works on Logger Pro set up to calculate the period of oscillations from the motion.
This apparatus was designed so the period of the system was measured in the logger pro in order to find an appropriate function to later predict the weight of an object using its period.
First, we started by measuring the period of the inertial balance on its own and later added 100 g weight increments until we reached 800 g. After gathering all this data we proceeded to make a new data graph filled with our measurements including the mass of balance and period. We also added three new columns, the masses of tray and weights, the natural log of T, and the natural log of the sum of the masses and tray.
The we created a plot and tried to narrow down the parameter Mtray until the correlation coefficient was as close to 0.9999 as possible. By exploring the different values of the Mtray we saw that there were a range of Mtray values that could yield the same correlations.
Data Tables:
These were our values in order to figure out what the graphs of our Mtray functions. We figured the highest correlations due to our guessed mass of Mtray and then we found the slope and y-value.
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| For example: One of the values here is 290g for the Mtray and the correlation is 0.9998 |
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| After we have our second function that has a Mtray value of 315 g and a correlation of 0.9998 |
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| In this graph we have the function of the mass of 340g and a correlation of 0.9998 |
Once we found the three different equations for the different values of Mtray, we record the period of two objects of unknown masses using the equations we derived.
These graphs have three added columns:
(Blue) the mass of the weights + the unknown mass of the tray
(Green) the natural log of period of oscillation
(Orange) the natural log of the masses added to the inertial balance
Apart from that we also have the threshold of our Mtray mass
Conclusion:
In order to find the mass of an object through its period was successfully proven with this experiment. By solving the formula of the natural log and equating it to equation of period we were able to create an equation in its line form. Graphing multiple points for our natural logs of its masses and periods resulted in a very close estimate of what the mass of an object is due to its period. Our results came fairly close to the actual weight of the iphone and calculator. Although our calculations were extremely close, we feel human error also played a small part in our experiment. Our mass calculations were off by about a gram from our first formula. This may be due to the placing of the tape that indicated the sensor of the oscillating motion. This may have resulted in an uneven reading thus creating some error in our calculations.
Most of the parts are here. Some things that would make this better, in no particular order:
ReplyDeletePart of the idea of measuring inertial mass is that gravity isn't involved in the measurement, so we aren't measuring the mass's weight. You mention weighing tow of three times right at the beginning of the blog.
You might simplify your description of the timing arrangements: A piece of tape on the end of the inertial pendulum passed back and forth through a photogate that was connected to LoggerPro. This arrangement allowed us to measure the period of oscillation.
There's no discussion of why you take the ln of both sides. You say "we created a plot" but not of what, or how that particular plot is going to get your to your results. You were adjusting the value of Mtray until you got your best straight line. When the correlation was as close to 1 as possible, that told you you had the best possible straight line. Since a range of values gives you this same best correlation, there is a range of possible values for A and n in your fit equation.
Somewhere you should post a table of your results for ln A and n (and A) that go with each value of Mtray from your graphs.
It might make more sense to have the table with the results of your calculations nearer the end, after you have derived your values of A and n.
The lab handout explicitly asked that you time with a stopwatch and record some known number of oscillations and compare that timing results with what you get from the photogate.
Consider other sources of uncertainty in the lab. All of the masses we used were cylinders, mostly at the center of the tray. The unknowns you tested had a different shape and might not have been centered on the tray. We didn't test to see if the shape or placement of the mass made a difference.