28-Aug-2014: Power law for an inertial balance.

The purpose for this experiment was to find a relationship between mass, of any object, and period for an inertial balance.

Below is the setup of the experiment. We started by clamping an inertial balance, a spring device which oscillates (side to side in this case), to the table top. At the outer end of the balance we placed a piece of tape. The piece of tape was adjusted so that it would pass through a photogate. Every oscillation cuts an invisible beam and after two full oscillations a period, length of time, is counted then recorded into Logger Pro on the lap tops.

Below you can see the data table where I wrote down the measured period for each of the different masses that were added to the end of the inertial balance. As we can see there seems to be a correlation between mass period but why has the larger mass resulted in longer period? Mass was recorded in grams and period in seconds.

The masses were converted to kilograms and the data was then plotted on a graph that resembled the standard linear graph of:

y = mx + b 

Our original equation looked like this:

T = (M+m)^n + A

Where M is the mass of the tray, and m is the known mass placed at the end. We then have our equation a bit of a modification in order to make it more pleasing to the eyes. We did so by taking the natural log of the whole equation and obtained:

lnT = (n)ln(M+m) + lnA

        Mass of the metal weights, M: KNOWN

        Mass of the tray: UNKNOWN

        Slope of the graph, n: KNOWN

        The y-intercept of the graph, lnA: KNOWN
        The y components, lnT or ln(period): KNOWN
        

Below you can see the chart I made where I included the known parts and included the correlation, which tells us how well the line and points fit together. We then tweaked the value of n to find a low and high value for M that would give us .9999 correlation.

        M low: 350g              M high: 440g

        lnA low: -0.4582        lnA high: -0.5273

        n low: -0.7163            n high: 0.8066

Finally, assuming that we ran the experiment correctly and collected the proper values we should be able to return to our equation of:

lnT = (n)ln(M+m) + lnA

and use it to calculate a very close approximation of an unknown mass at the end of the inertial balance. Below is an other table where I ran the experiment all over again using a laptop charger and a wooden block, both of unknown mass, and wrote down the period, T, of each.

        Mass of charger: unkown        Mass of wood block: unknown

        T = .432 seconds                    T = .401 seconds

Using the data from the previous two tables we can calculate a range for the unknown masses, which I've done below. 

From all of this we are able to derive a fairly accurate power law.