25 and 27 - August - 2014: Deriving a power law for an inertial pendulum

 Deriving a Power Law from an Inertial Pendulum

       In this lab we tried to find a relationship between mass and period for an inertial balance by 

comparing objects resistances to their changes in their motion.

Lab Equipment   

     The picture above contains the setup of an inertial balance along with weights, added later, and a

photo gate. These are used to find a relationship between the period of time that it takes an object of

some mass to pass through the photo gate twice.

From Data to Fulling Purpose

     After getting the data we plugged it into Logger Pro where we graphed the data. The objective was 

to find three unknowns in the equation T =A(m +Mtray)n so that we would be able to find the mass of 

two unknowns, given their period on the inertia pendulum. The three unknowns were A, Mtray, and 

n. We were able to do this by taking the natural log of each side lnT = n ln (m +Mtray) + ln A. We 

found, from plugging in the numbers, that the mass of the tray was in between 0.28kg and 0.33kg 

coming to an mean of 0.305kg. As this range for the mass of the tray created a beautiful straight line 

as seen by the very strong correlation between the plotted points. The slope of the line from the graph 

was n, the range was in between 0.6524 and 0.7133 having a mean of 0.68285. We found lnA to be 

-0.4168 and -0.4479 having the average of -.43235. Next from lnA we found the final unknown A to 

be in between 0.63897 and 0.65915 having the average of 0.649035  After we found that the three 

unknowns from the original formula we each separately plugged in several of the times we found 

from the known masses to confirmed that the formula and numbers closely matched up. As seen

below.

       In the lab we also found the periods of two unknown masses. Due to error on my part I do not 

have their times just an approximation based from the formulas. This is due to me being able to 

closely measure the masses of the two unknown masses, my calculator and a heavy duty carabiner 

having masses of approximately 0.292 kg and 0.105 kg from a scale in my home. The resulting 

periods were for my Unknown 1 with the mass of 0.292 kg being 0.4563 sec and for my Unknown 2 

with the mass of 0.105 kg being 0.353 sec. The math is seen below.

The Data Table is shown below

A screenshot from Logger Pro mass of the tray is 0.280 kg

     In this screenshot above the fit line is the lowest range at 0.280 kg of the mass of the tray creates 

a line with a very strong correlation. between the natural log of the mass of the tray and the mass 

of the object against the log of the period of motion.

Another screenshot from Logger Pro mass of the tray is 0.330 kg

   

  In this screenshot above the fit line is the lowest range at 0.330 kg of the mass of the tray creates a 

similar line with a very strong correlation. between the natural log of the mass of the tray and the 

mass of the object against the log of the period of motion.

Summary

     In this lab as a series of groups we attempted to find the relationship between mass and period by 

using an initial balance. As individual groups we took data using a piece of tape connected to an 

inertial pendulum that would pass through a photo gate and record data accurately. We then used

that data to confirm that the formula T =A(m +Mtray)n given to us was able to closely match the 

numbers we collected in our data. This allowed us to find two unknown masses from two objects 

of different masses given their periods of time passing through the photo gate. Though I had to 

find the time due to my misplacing of the information I took down.

Thank you for reading my lab blog.