23-Feb-2015: Deriving a power law for an inertial pendulum

Purpose of Lab: To find the mass of an object using the relationship between mass and period of an object on an inertial balance.

Tools:

• Inertial balance/tray
• C-clamp
• Various amounts of weights
• Photogate

Apparatus used to measure pendulum oscillation.

Procedure/Theory:

  We set up an inertial balance at the edge of a table that would pass through a photogate, measuring the period of one full oscillation.  We performed trials with varying values of masses each time, which were 0-800 gram weights in increments of 100, a 180.2 gram wooden block, and a 600.5 gram tape dispenser.  Each yielded a different period of oscillation.  When forming a graph (excluding the wooden block and tape dispenser), we also included an added assumed mass of 150 g for the tray (balance) and got the equation T=A(mass+Mtray)^n.  We took the log of each side to form a linear equation, and then plugged in different values of Mtray to determine what y-intercept and slope values would yield the best correlation.  This gave us a lower and upper limit for Mtray, and we used that to verify the values of the two objects (wooden block and tape dispenser). 

Data Table. Not pictured:
Wood Block (mass 180.2 g, period .394 sec) &
Tape Dispenser (mass 600.5 g, period .594 sec)

Data Table, including mass of tray and log of period and mass.

Non-linear relationship between mass and period.

Linearization of previous data, with graph of best fit line.

We changed the mass of the tray for equation T=A(m+mtray)^n, which gave two parameters for a best fit correlation of the line:

Mass of tray at 260 grams.

Mass of tray at 300 grams.

When Mtray was set at 260 grams, the value of A was e^-4.687 and the value of n was 0.616 in the equation T=A(m+mtray)^n.

When Mtray was set at 300 grams, the value of A was e^-5.043 and the value of n was 0.665 in the equation T=A(m+mtray)^n.

Calculations to verify what the mass of each object was using the masses of trays at best fit linear correlations.

Results:  We found that the best correlation was achieved when we set Mtray at a minimum of 260 grams and a maximum of 300 grams.  For the wooden block, it yielded a mass of 183.09 to 185.72 grams, where the actual mass was 180.2 grams.  For the tape dispenser, it yielded a mass of 602.55 to 600.78 grams, where the actual mass was 600.5 grams.

Conclusion:  The most accurate result when determining the mass of the objects showed us that the tray is somewhere between 260 and 300 grams.  In addition, we proved that the mass of objects can be determined through inertial balance.

Uncertainties/Assumptions:

• The placement of various masses on the tray could have been slightly different.
• The precise mass of the two objects could have been measured incorrectly.
• Amount of tape/how we taped the weights and objects on the tray may have altered oscillations.
• The distance we pulled the tray back for each period measurement may have been slightly different.
• The number of periods we let the tray oscillate for each time may have been different.