The purpose of this lab was to experimentally investigate and find the relationship between the period and mass of an inertial pendulum. This relationship is found in the derivation of an equation that relates the two variables, period and mass.
Apparatus:
This apparatus consists of an inertial pendulum that oscillates horizontally back and forth. What is unique about this kind of pendulum is that gravity has no effect on it. The inertial pendulum was used to experimentally investigate the relationship between the mass added and the time it requires for the pendulum to oscillate through one cycle, known as the period. The measurement of the different periods that correlated with the different amounts of masses added was done via the photogate on the tip of the inertial pendulum. The photogate contains a vertical beam that, when disrupted or blocked, counts as half an oscillation. When the beam is distrupted twice, it counts the time taken to do so as one period. A thin piece of tape was attached to the end of the pendulum in order to measure period by having the beam not blocked by the actual plate of the pendulum the whole time. The data for the measurements of period were taken via LoggerPro, a computer software.
Abstract:
The above graph displays the data found in the data table above, with the mass added as the x axis and the period corresponding to the mass added on the y axis. As is seen in the graph, the relationship between the mass added and the period of the inertial pendulum is not proportional. Therefore, it was guessed that the relationship of T (period) and m (mass added) was by the power law T = A(m + mass of tray)^n, where A and n are unknown constants. The mass of the tray was initially estimated to be 0.30 kg. Taking the natural log of both sides, the formula ln(T) = n ln(m + mass of tray) + ln(A) was derived. This formula is in the form y = mx + b, where n is the slope and ln(A) is the y-intercept.
Using the data in the data table above, the graphs above were constructed using the equation ln(T) = n ln(m + Mtray) + ln(A). The linear fit in each graph shows how close the guess of the mass of the tray is to the actual amount. The linear fit obtained that was closest to 1 was 0.9996. The value of this linear fit was conserved when the mass of the tray was adjusted from the lowest value of 0.28 kg to the highest value of 0.36 kg. The higher graph above shows the slope, y-intercept, and linear fit for the lowest value of Mtray (0.28 kg), whereas the lower graph above shows the slope, y-intercept, and linear fit for the highest value of Mtray (0.36 kg). Slope and y-intercept are of much importance because they can be used to solve for our unknowns, A and n. Looking back at the formula ln(T) = n ln(m + Mtray) + ln(A), A can be found for the highest and lowest values of Mtray by taking e to the power of the y-intercept (ln A), and n can be found by recording the slope of the graph. Once the values of A and n are found, the formulas that relate period and mass of an inertial balance for highest and lowest values of Mtray can be found. The derivation for the two formulas are found in the photograph below, where T = A(m + Mtray)^n is the general equation.
To ensure that these two formulas are indeed the correct models for the behavior of the inertial balance, they were tested by measuring the period of the pendulum using two unknown added masses. The smaller added mass was a cardboard box (172 g), and the period measured with it as the added mass was 0.3919 seconds. The larger added mass was a mostly filled water bottle (601 g), and the period was measured to be 0.6152 seconds when it was the added mass. Using the measured periods for both added masses, the range for the expected values of the added masses were found by plugging the measured periods in the above equations that relate mass and period, as seen in the pictures below.
In the calculations, below, the measured periods of the unknowns were plugged into the formula ln(T) = n ln(m + Mtray) + ln(A) instead of T = A(m + mass of tray)^n. However, both formulas are equivalent and are equally useful in solving for the ranges of the expected masses of the two unknowns.
This picture contains the calculations for the range of mass values for the attached cardboard box. The range calculated was from 0.0992 kg to 0.2660 kg, The measured mass of 0.172 kg is located in that range, which is strong proof that these equations are correct models.
This photograph contains the calculations for the range of mass values for the attached mostly filled water bottle. The range calculated was from 0.5624 kg to 0.7223 kg, The measured mass of 0.601 kg is located in that range, which is additional strong proof that these equations are correct models.
It was found that the measured values of the masses of the two "unknown" attached masses were in the range of the calculated low and high values through the use of the derived formulas that relate mass and period of the inertial balance. Since the measured values and in the range of the calculated ones, it is strong proof that these derived formulas are accurate representations or models of the behavior of the inertial pendulum.
Procedure:
The apparatus shown in the photo above was first set up by using a C-clamp to hold down the inertial pendulum, tape to hold down the added masses and to act as the "blocker" of the photogate beam, a photogate to measure the period of the oscillation of the pendulum using LoggerPro, and a computer with LoggerPro and the photogate attached to. After, the period of the tray with no added mass was first measured, and then the period was measured with a 100 g mass attached to the pendulum. This was repeated again 7 times, with each time adding an extra 100 g mass. In the final trial 800 g of total mass was added to the pendulum.
Later, the same procedure was done two more times, with the only difference that a cardboard block and a mostly filled water bottle were used as added masses. The cardboard box had a mass of 172 g and the period when it was the attached mass was measured to be 0.3919 s. The filled water bottle had a mass of 601 g and the period when it was the added mass was found to be 0.6152 g.
The graph of mass (kg) vs period (s) was made using the data from the data table and the graphs of ln (m + Mtray) vs ln(T) was made using the data from the same data table but plugged into the derived formula ln(T) = n ln(m + mass of tray) + ln(A), where the range for the experimental mass of the tray that was closest to the actual mass of the tray was found to be from 0.28 kg to 0.36 kg, based on the conservation of the highest correlation coefficient of 0.9996. Using the graphs of ln(m + Mtray) vs ln(T), the unknown values of A and n were found for the lowest and highest values of Mtray.
Once the formulas for the highest and lowest value of Mtray were derived, they were tested by seeing if the calculated range of the highest and lowest values for the masses of the "unknowns" included the measured mass of the attached "unknown" mass, which was found to be true.
No comments:
Post a Comment