23-Feb-2015: Using a derived power law to determine unknown masses using an inertial balance

Inertial Balance Lab

(measuring mass without depending upon gravity)

• Purpose:  Commonly, the mass of an object is measured utilizing the acceleration due to gravity acting on the object.  It is known however, that the mass of an object is constant, regardless of how the Earth's gravitational force is affecting it.  This lab demonstrates how to measure the mass of an object utilizing a device known as an Inertial Balance.  This device is used to measure the inertial mass of objects by way of comparing objects' resistances to changes in motion.

• Experiment Equipment:

(Complete apparatus set-up)

The red & black object is the Inertial Balance device described in the purpose section above.  It is held in place at the edge of the table via a C-clamp.  Attached to the rod in the photo, is an oscillation period measuring device known as a Photogate.  This device consists of an infrared diode and a photocell.  Timing on this device occurs when the infrared beam is interrupted; to cause this interruption, we placed a small strip of masking tape at the tip of the inertial balance (opposite the table).  The photogate will record a period each time the beam has been interrupted 3 times (which constituted an entire period of an oscillation).

• Data Collection:  The photogate (depicted above) is connected via a Logger Pro device to a MAC computer system.  The Logger Pro software is used to attain the average period of the oscillatory motion recorded by the photogate.  We began with recording the average period with no mass (aside from the tray's own mass) placed on the inertial balance.  We then incrementally increased the mass by adding 100g each time, up to a maximum of 800g.  This is done in order to "calibrate" the inertial balance system.  The recorded data is depicted below.

(Period values recorded via Photogate device)

Also recorded (not depicted) were two additional objects.  A stapler, and a calculator.  Their information is given below:

      Stapler:

           Mass (using standard scale) = 369 g

           Period of oscillation (via photogate data) = 0.495s

     Calculator:

          Mass = 148g

          Period of oscillation = 0.376s

• Interpreting the Data:  We will guess that the period and mass are related by some power-law equation, given as :

                            T = A(Ma + Mtray)^n

    

          Taking the natural log of both sides yields:

                           lnT = n*ln(Ma + Mtray) + lnA

          This second equation resembles that of y = mx + b.  

          This similarity implies that n = the slope of the plotted line, and that lnA = (y-int).

The mass of the tray (Mtray) is attained by adjusting the parameter (created in Logger Pro) until the linear fit yields a correlation coefficient as close to 1 as possible (typically 0.99xx).  This can occur over a range of values entered into the parameter.  In our case, the minimum and maximum values for Mtray which gave best correlation values, which in turn gave excellent curve fits were 280g and 300g respectively.

          Now, when the value of Mtray is at it's minimum, the following data is presented:

                 Mtray = 280g

                 Slope = m = n = 0.6489

                 Y-int = -4.905

           Value of Mtray at maximum:

                Mtray = 300g

                Slope = m = n = 0.6733

                Y-int = -5.085

We now have all data necessary to calculate the inertial mass of the stapler and calculator to compare against their recorded masses using a traditional scale!!!

          Solving the equation lnT = n*ln(Ma + Mtray) + lnA for Ma yields the following:

          Ma = e^[ (lnT - lnA) / n ] - Mtray

We use the preceding equation to determine the maximum and minimum values for the stapler/calculator masses using the maximum and minimum values of Mtray, n, and y-int!!!

• Results:  

(Derivation of Ma equation (top); range of values for "unknown" masses (bottom))

We now have a range of values associated with the masses of the stapler and calculator, they're given as follows:

                Stapler: 370.45 g - 374.76 g                Measured Mass = 369 g

                Calculator: 144.76 g - 145.66 g          Measured Mass = 148 g

It appears there is in fact some uncertainty in our calculations, nevertheless, our estimates are quite close to the measured values.

• Summary:  The overall purpose of this lab was to utilize a method of measuring an objects mass essentially in a vacuum (ignoring the Earth's gravitational force).  This is useful, for example, in space, where the acceleration due to gravity is absent; but just because gravity is absent, does NOT mean that objects are mass-less!  The methods used in this lab undoubtedly prove this.