Lab 1: Inertial Balance

Jin Im

with Victoria Bravo and Jonathan Goei

27 February, 2017

By using a photogate, series of masses, and an inertial pendulum, we sought the relationship between mass and the period of said pendulum. Once we found this relationship and derive the equation, we measured the period of objects with an unknown mass on the pendulum in order to predict its mass. By graphing the periods of multiple masses on the inertial balance using the given formula T=A(m+M)^n, we seek to find the degree of correlation between period and mass.

The experiment was set up such that the inertial balance, which has a horizontal degree of motion, was attached to the edge of a table. A photogate connected to a laptop, which we used to measure period of oscillations, was placed directly opposite the balance. A thin piece of tape as affixed to the end of the balance, as thinner objects passing through the photogate give us a more accurate period. The apparatus is shown below. 

When an object with mass m is placed onto the stand, as shown, and the system allowed to oscillate, the laptop provided a period T. We replicated this 9 times with different values of m: 0g, 100g, 200g, 300g, 400g, ... etc. 

Using the given formula T=A(m+M)^n, in which M is the mass of the tray oscillating with the masses and A and n are constants, we see that by using different values of m and their corresponding T, we can predict the other values. The formula above was converted to slope-intercept form by taking the natural log of both sides: ln T = n ln(m+M) + ln A. Ln(m+M) becomes our independent variable, ln T becomes our dependent variable, n becomes our slope, and ln A becomes our y-intercept. 

Although the value of M was unknown, we were able to predict it by using a dynamic variable for M in plotting ln T = n ln(m+M) + ln A. By gradually 'sliding' the value of M in the graph, we were able to find the value of M such that the graph had formed a straight line with a correlation coefficient close to 1. Our experimental value for M ranged from 282g to 351g.

Data Table assuming M is 282g

ln T = n ln(m+M) + ln A when M is 282g

282g is the first value for M such that the correlation value on the graph reaches 0.9998,
an acceptable value for the purposes of this lab.

when M is 300g

when M is 351g

These graphs include everything that we need to know to describe the relationship between the mass

and period. Looking at the formula T = A(m+M)^n, we see that n is the slope of the lines of the graphs

above. ln A is the y-intercepts of these lines (A can also be found by T = A(m+M)^n as we now have 

all the variables). The data chart of all relevant values at differing values of M is as shown: 

          For example: T = A(m+M)^n

                                 0.503 = A(400+300)^0.689

                                 0.503 = 91.26A

                                 A = .0053

With these values, we were also able to predict the mass of certain objects by placing them on the apparatus. We used two objects: a small basketball and my TI-83 calculator.

For example, the basketball:

We measured the period of oscillation to be 0.45798 seconds.

0.45798 = .0065(m+ 282g)^0.6695

70.15 = (m+282g)^0.6695

571.9 = m + 282

m = 289.0 grams, assuming M is 282g

0.45798 = .0035(m+351)^0.7499

130.85 = (m+351)^0.7499

664.9 = m+ 351

m = 313.9 grams, when M is 351g 

Our range of mass of the basketball is between 289.0g and 313.9g. The actual value of the ball, as measured on a scale, turned out to be 291g.

The period of oscillation for the calculator was 0.4348s, and the range of mass was from 250g to 269g, while the actual value was 255g. 

Conclusion

The experiment to find the relationship between the mass and period of an object was a roundabout one, with variable values with a degree of uncertainty. Because we had to rely on a range of values for the mass of the tray, the relationship between the mass and period of a inertial pendulum we derived from this experiment is ranged as well. We also performed the experiment with the assumption that the tape attached to the tray of the inertial balance would not sway in the oscillations, which perhaps yielded inaccurate period readings.