23 February 2015. Deriving a power law for an inertial pendulum.

Purpose
to find a relationship between mass and period for an inertial balance. After the relationship is modeled by an equation, individuals can find the mass of the unknown objects by measuring their period by the inertial balance device.


Apparatus:

1. C- Clamp: c-clamp is a device that is used to secure object between the flat end of the screw and the flat end of the frame.
2. Photogate: A photogate sensor is a timing device used for very precise measurements of high-speed or short-duration events. It detects an object which will move through and blocks the beam of light between the source and the detector.
3. Logger pro: A set of devices and software that can analysis data taken from an experiment.
4. Inertial balance: a device that acts and helps objects to oscillate inertially.
   

Procedure:

1. Set up the photogate, c-clamp, and the inertial balance as shown in the picture, such that when the balance is oscillating, the tape completely passes through the beam of the photogate.

     2.  Connect the photogate to the logger pro (open application pendulum timer).

     3.  Try some oscillations then record the time with a stopwatch to make sure that the application works.

     4.  Record the period without any mass in the tray, and vary it from 100g-800g (adding 100g each time).

Data analysis:

• We know from the experiment that the graph with mass in comparison to period is as follow:

• From the graph we guess that the period is related to mass by some power-law time of equation which can be denoted by:

T=A(M_Tray+M_Added)ⁿ

• From this equation, we have three "unknowns"; A, M_Tray, and n. We have to determine the values and make a perfect guess for the mass of the tray, by taking natural logarithm of each side >>     

lnT = n ln(M_Tray+M_Added) + lnA, which looks like Y = mx + c

• Therefore, we need to plot lnT vs ln(M_Tray+M_Added) which will gives us a straight line (if we guess the mass of the tray perfectly), with the slope n and y-intercept lnA. (We can adjust the value of the mass of the tray by going to the userparameter menu in LoggerPro).
• In our experiment, we find the nearest-to-1 correlation coefficient we can get is 0.9998, with the mass of the tray equals from the range 0.253 kg until 0.295 kg.

Calculation

• Here is the graph with the higher mass of the tray = 0.295 kg, with slope = 0.6650 and y-intercept = -0.3971. (correlation = 0.9998)

• Here is the graph with the lower mass of the tray = 0.253 kg, with slope = 0.6213 and y-intercept = -0.3718. (correlation = 0.9998)

• Calculation done in the picture as shown:

EXTENSION

• From the calculation above, we got our two models of equation. Therefore we can use the equation to determine the mass of the unknown objects by measuring their periods of oscillation. In our group, we use calculator and tape as the objects.
• Remember, because of the uncertainties in the mass of the tray before, we found two models of equation, meaning that the mass of the calculator and tape will not be exact, but they are going to be presented in range.

• We measured and found out that the period of oscillation of the calculator is: T=0.4742 s, and the tape T=0.382198 s by using the inertial balance device and logger pro.
• From the equation that we modeled, now we can calculate the mass, as shown in the picture below.

• As a comparison, we compare the masses found from the equation with the direct measurement of the calculator and object on the digital balance.

• We can see that the mass of calculator is off by: 0.308kg-0.297kg = 0.011 kg. The error may occur due to some external factors such as:
• the inconsistency of the force of oscillation. The differences of magnitude of force to make the inertial balance oscillate (with or without mass) will produce different results.
• rounding figures while calculating. Due to some complicated calculations, the amount of significant figures are reduced to simplify equation. This would result into a slight differences in the outcome of the final result.
• other non-calculated factors.
• While the mass of the tape is off by: 0.134kg-0.134kg = 0 kg, which means that the model of equation is accurate this time.

Conclusion

Therefore, the relationship between mass and period for an inertial balance is modeled by the range of equations of:

• For higher Mtray, T = 0.672268 (M_{Added +} .295) ^0.6650
• For lower Mtray, T = 0.6894921 (M_{Added +} .253) ^0.6213