For our lab, we were tasked with finding out the relationship between an inertial pendulum (seen above) and its mass. First, we set up the inertial pendulum by clamping it to the table without any weights on it. Next we got our laptop and LabPro all set up and connected with a photogate (seen above) plugged into the DIG/SONIC 1 input slot. The photogate takes data when something breaks through the beam so we attached a piece of tape on the end of the pendulum to pass through it. Theres a file in the computer called Pendulum Timer.cmbl that works through Logger Pro that you have to use with the photogate.
Next step is to test your photogate with a trial run. Time a number of oscillations and see if the numbers reasonably match up. Now you can start your trials starting with no weight on the end and working your way up to 800 grams in increments of 100 grams. Seen below are my trial periods.
After you get your trial period times, find the periods of two mystery items. My group used a black tape dispenser=0.605kg, and one of our group's keys=0.105kg. We now guess that the period is related to the mass by a power-law type of equation T=A(m+Mtray)^n. Unfortunately we have three unknowns in the equation. In order to solve them, we are going to change the equation into something we are used to working with. Taking the natural log of both sides of the equation gives us lnT=n*ln(m+Mtray)+lnA which resembles the equation of a line y=mx+b.
Now input the masses and periods into the rows and columns of a new Logger Pro file only after you close the photogate file. Go to the Data drop down menu and click on Column Options. There you can designate X and Y to be Mass in kg and Period in seconds. Now go to the Data drop down menu and click on User Parameters to create the parameter Mtray and assign it a weight with an educated guess. Once again go to the Data drop down menu and this time click on the New Calculated Column and create three new columns for m+Mtray, lnT and ln(m+Mtray).
Now plot lnT vs ln(m+Mtray) and apply a linear fit to the points. The line that says correlation is how close your points come to making a straight line and is also dependent on how close your educated guess was to the actual mass of the pendulum. Now you just need to play with the mass values until you get a range of masses that gets you the correlation closest to one. For us, the range of mass values that got us the highest correlations was from 0.295g-0.300g. From the information in the box, you can substitute in the numbers you get into your power-law equation. From the graph (seen below) m=0.6664 and our b=-0.429. After plugging in our values from the graph into the formula, I took the inverse of the natural log of both sides of the equation and plugged in numbers to get values for the period (example below).
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