11 March 2015: Propagated Uncertainty in Measurements

Purpose:

The purpose of this experiment was to become more aware of uncertainty and its importance by calculating the propagated uncertainty in measurements of density and forces on a static system.

Apparatus:

                                                   Figure 1: Photograph of the vernier caliper

The first portion of the experiment involved finding the densities of three metal cylinders with differing dimensions and different types of metals. A vernier caliper was used to measure the dimensions of the cylinders. A vernier caliper was used instead of a ruler because it is more accurate, decreasing uncertainty in the measurements of dimension. The uncertainty of a caliper is 0.01 cm, whereas the uncertainty of a ruler is 0.1 cm. A vernier caliper, as shown in the figure below, contains an additional 0.01 cm marking. When a measurement is performed with a caliper, it is read by looking for the 0.01 cm marking that best aligns with one of the 0.1 cm markings. That 0.01 cm marking is the hundredths place value for the measurement. This is why the uncertainty in a vernier caliper is one digit less than a ruler.

                                         Figure 2: Photograph of 0.01 cm markings on the caliper

The vernier caliper works by adjusting the length between its "arms" to fit the dimension of the object that is measured. Doing so gives no uncertainty for the tenths place but only for the hundredths place.

                                Figure 3: Photograph of magnetic protractor

The second portion of the experiment involved determining an unknown hanging mass by measuring the forces that were acting on the mass. The magnetic protractor, shown above, was used to measure the angle of the attached ropes (force vectors) from the horizontal. This apparatus is used by placing it parallel to the slope of the interest, which then gives the reading of the angle of the slope.

Procedure (Part 1):

    Figure 4: Measurements of the dimensions and masses of the three cylinders

Using the vernier caliper, the dimensions of each of the above cylinders were measured and recorded. In addition, the masses of these cylinders were measured using a balance that has an uncertainty of 0.1 g. Once the dimensions were measured, the density of each cylinder was calculated. The top cylinder was numbered 1, the middle cylinder 2, and the bottom cylinder 3. Cylinder 1 contains a density of 4.26 g/cm³. The density of cylinder 2 was calculated to be 2.89 g/cm³. Lastly, the density of cylinder 3 was found to be 9.22 g/cm³.

    Figure 5: Diagram to solve for the volume and density of a cylinder

The volume of the cylinder was found by squaring half the diameter of the cylinder, followed by multiplying the height of the cylinder and the value of pi, as seen in the above figure. Then, the density was found by taking the measured mass, in grams, and dividing it by the volume of the cylinder, as seen above. 

Abstract (Part 1):

    Figure 6: Uncertainty in density partial derivative equation

The uncertainty in density was found by taking the partial derivatives of all the variables of density, then multiplying them by the their uncertainty and summing them together (as seen above). Since density is a function of 3 variables: m, d, and h ( where m = mass, h = height, and d = diameter), three partial derivatives each with respect to one variable was made. A partial derivative is taken by treating the variable it is with respect to as a variable and everything else (including other variables) as a constant. Doing so gives you the three partial derivative equations shown in the picture below:

    Figure 7: Partial derivatives of density with respect to each of its variables

Using these equations and plugging their values into the equation shown in Figure 6, the uncertainty of density can be found. The uncertainty in the measurement of mass (dm) is 0.1 g, whereas the uncertainties in the measurements of height and diameter (dh and dd, respectively) are 0.01 cm.

    Figure 8: Finding the uncertainty in density of the first cylinder

Figure 9: The densities and uncertainties in density of cylinders 1 and 2 and finding the uncertainty in density of cylinder 2

Figure 10: Finding the uncertainty in density of cylinder 3 and the values of density and uncertainty in density of cylinder 3.

Using the values of the densities and their uncertainties of the cylinders, the identity of the metal of cylinder 2 was found to be Aluminum (d = 2.70 g/cm^3). The metal that makes up cylinder 3 was identified to be copper (d = 8.96 g/cm^3). The identity of the metal of cylinder 1 was not successfully identified due to having a small density yet a color different than the lighter metals. Some possibilities that could have caused the calculated densities to not be as close to the densities of metals include:

1) The cylinders were alloys (a mixture of two metals) yet were assumed to be only one metal and compared to pure metals

2) Rusting of the metals might have caused their mass to increase, resulting in a larger than actual      density.

3) The cylinders might not be completely solid on the inside (i.e. the cylinders have some empty    space inside), reducing the mass and therefore obtaining a smaller density.  

The uncertainties of the densities ranged from 1% to 2% of the density values, which correlated strongly with the uncertainties in the used instruments.

Procedure (Part 2):

The setups shown below in Figures 11 and 12 were already present before this experiment. The tensions on each of the strings in each setup (displayed by the 10 N spring scales) were measured as well as the angles of the strings from the horizontal. The instrument used for the measurements of the angles was the magnetic protractor shown in Figure 3. The uncertainty in the angle measurements is 2 degrees, and the uncertainty in the tension measurements in 0.5 N. 

    Figure 11: Setup for hanging unknown mass number 1

For the setup of unknown mass # 1, the tension in the left string was recorded to be 4.8 N, and the tension in the right string was 6.4 N. The angle of the left string was measured as 26 degrees and the angle of the right string was measured to be 46 degrees.

  Figure 12: Setup for hanging unknown mass number 8

For the setup of unknown mass # 8, the tension in the left string was recorded to be 10.5 N, and the tension in the right string was 7.2 N. The angle of the left string was measured as 48 degrees and the angle of the right string was measured to be 10 degrees.

Abstract (Part 2):

Once all of the tensions and angles were measured, free body diagrams were drawn for each of these two setups, as shown below:

Figure 13: Free body diagrams, the mass equation, and solving for the masses of the unknowns in   the above two setups

Summing the forces in the y-direction, the unknown masses can be found as a function of the two tensions, the angles, and gravity. The x-direction need not be summed since there is no unknown variable or mass component in the x-direction. Used the derived formula above for mass, the unknown masses were calculated.

However, since there was uncertainty in the measurements of the tensions and angles, there must be uncertainty in the calculated masses. We can find the uncertainty by taking partial derivatives of the above derived equation for mass with respect to each of the 4 variables: The tension of the first string, the tension of the second string, the angle of the first string, and the angle of the second string. Once the partial derivatives with respect to each variable were found, they can be multiplied each by the uncertainty of the measurement itself (i.e. 2 degrees (in radians) for angle and 0.5 N for tension). The degrees must be changed to radians since partial derivatives is a calculus operation. The partial derivatives are shown in the upper part of the figure below:

Figure 14: Derived partial derivative equations for the uncertainty in the calculated unknown masses / solving for uncertainty in unknown mass # 1

  Using these partial derivative equations, the uncertainties in the masses of the unknowns of the two hanging mass setups were calculated by plugging the values into the formulas, then summing up the value for each of the four formulas. The uncertainty in the measurements of the angles and tension forces are already included in the partial derivative equations (dF and dθ).

Figure 15: Solving for uncertainty in unknown mass number 8

 The uncertainties in the masses ranged from 10% to 15% of the calculated masses, which was expected since the uncertainties in the instruments  used (added up) were roughly in the same range.

Conclusion:

The purpose of this experiment was achieved in that the importance of taking into account uncertainty of any measurements was learned. In addition, the propagated uncertainties of the densities and unknown masses were successfully calculated. Most, if not all, of the uncertainties of this experiment were found in the uncertainties of the instruments and the measurements of masses, tensions, and dimensions. Most of the error in this experiment was found in the uncertainties of the instruments and measurements, especially in the second portion of the experiment. One major error in the first portion was the assumption that the cylinders were fully solid, had the shape of a perfect cylinder, and were composed of pure metal of one kind. The identity of the metal that composed the first cylinder was not found due to the density being small yet the qualitative features being different than the lighter metals. The identity of the second cylinder was found to be aluminum, and the identity of the third cylinder was found to be copper.

4 March 2015: Non-constant acceleration activity

Purpose:

The purpose of this lab was to solve a non-constant acceleration problem numerically using Excel. Solving the problem numerically was much faster and easier than solving it analytically.

Apparatus:

    Picture 1: The non-constant acceleration problem solved in this activity

The apparatus of this lab  consisted solely of a computer with the Excel program and the problem shown in the picture above. The information given from the problem is the initial velocity of the elephant (25 m/s), the final velocity of the elephant (0 m/s), the force going in the opposite direction of velocity caused by the thrust of the rocket (-8000 N), and the mass function that models the change in mass of the rocket strapped onto the elephant: m(t) = 1500 kg - (20 kg/s) t. 

Abstract:

In order to solve this problem analytically (by hand), the mass function that models the whole system being studied must be found. In this example, the whole system is the elephant as well as the rocket. The mass function of the rocket is given explicitly but the elephant's is not. Conceptually, though, it is known that the mass of the elephant does not change; therefore, the mass function of the elephant is just: m(t) = 5000 kg. Now that we have both the mass function of the rocket and elephant, we can find the mass function of the whole system by just adding the two to get: m(t) = 6500 kg - (20 kg/s) t.   

Now that the mass function is known, we can find the acceleration function of the system by looking at Newton's Second Law of Motion, F = ma. For our purposes, we can rearrange the formula to find acceleration (a = F/m). Using the constant force of 8000 N and the mass function, we find that for our system: a(t) = [(-8000 N) / (6500 kg - (20 kg/s) t)].

Integrating the acceleration function from 0 to t we get our change in velocity (delta v). Our velocity function is then found by adding the initial velocity to the change in velocity, as seen in this equation: v(t) = Vo + delta v. Setting the velocity function equal to zero, the time it takes for the elephant to come to rest can be found. The time was calculated to be close to 19.69 seconds.

Lastly, integrating the velocity function gives us the change in position. The position function can then be found by adding the initial position (which we take to be the origin at the bottom of the hill) and the change in position (x(t) = Xo + delta x). The final position of the elephant can then be found by plugging in the previously found value of time in the position function. The answer calculate should be about 248.7 m. 

As may be seen, the integration required in this problem is complex and requires many techniques that are time consuming and increase the chance for error. Therefore, this problem was solved numerically using Excel.

Procedure:

    Picture 2: Table of model for non-constant acceleration problem

Column A in a new Excel spreadsheet was labeled t for time. The change in time for each row was 0.01 s. Columm B was labeled a for acceleration. Column C was labeled a_avg for average acceleration. Column D was labeled "delta" v for change in velocity. Column E was labeled v for velocity. Column F was labeled v_avg for average velocity. Column G was labeled "delta" x for change in position. Lastly, column H was labeled x for position with respect to the origin.

    Picture 3: Diagram for finding the values for each column in the Excel spreadsheet

The acceleration in column B was found by typing in the acceleration function previously found (-400/(325-t)) and plugging in the value for time at that instant (i.e. in that row). Average acceleration in column C was then found by taking the average of the acceleration value in that row as well as the row above. So for example, the average acceleration in cell C4 was found by taking the average of the acceleration values in cells B3 and B4. Next, the change in velocity in column D was found by multiplying the average acceleration by the change in time, which was 0.01 s each time. Cell E3 was value 25 since the initial velocity was 25 m/s. The rest of the values for velocity in column E were calculated by adding the change in velocity of that row with the velocity in the previous row. So, for example, the value in cell E4 was found by adding the values in cells D4 and E3. The velocity decreases as time increases because the system is decelerating (the rocket thrust is in the opposite direction of motion). The average velocity in column F was found by taking the average of the velocity at that instant as well as the velocity from the previous instant (just like how the average acceleration was found). The change in position in column G was then found by multiplying the average velocity with the change in time, which is 0.01 s. Lastly, position in column H  was found by adding the change in position at that instant with the previous position (the same way it was done for velocity in column E).

Discussion:

As is known, the change in velocity is the integral of acceleration and the change in position is the integral of velocity. In addition, integration provides you with the area under a portion of the curve of a graph. The size of  the portion depends on the limits of integration; a larger limit interval gives a larger area under the curve and vise versa. This was the underlying concept of the solution to this problem analytically.

However, integration cannot be done in Excel. Therefore, another way of finding velocity for acceleration and position from velocity is by manually finding the areas under the graph curves of acceleration and velocity. This method works perfectly when the curves have a constant slope i.e. acceleration or change in velocity is constant. However, problems are faced when the acceleration  is not constant i.e. the slopes of the curves are not constant, as in this problem. This is because it is difficult to calculate the area of irregular shapes under the curve.

However, a common solution to this problem is by looking at very small intervals on the graph. When the interval are "small enough", the curve of the graph being studied in the interval actually becomes straight. This can be seen when you zoom in very closely at a curve in a graph, as seen below:

    Picture 4: graph of y = x^2 zoomed in very close

This would take too long to be done by hand, which is integration is the best solution by hand. However, with a program like Excel, it is possible to look at very small intervals of the curve of a graph in a reasonable amount of time. With the intervals being small enough, the area under the curve of the graph can be calculate by looking at the average of the points that enclose the interval. This cannot be done with a large interval because the average is not an accurate representation of the slope of the graph. However, with very small intervals, the average becomes a very accurate representation of the slope of the curve in that interval, and the area under the curve can be found much more accurately and with much smaller error. This is the underlying reason for choosing a very small change in time for the Excel spreadsheet. This is proven when we look at the value obtained for the position of the system at 19.69 seconds highlighted below:

    Picture 5: The change in position and position from the origin of the system at 19.69 seconds

The value found for the position at 19.69 seconds (248.698165 m) using the area under the curve method is very accurate and almost equal to the value found via the integration technique (248.698166 m). This proves that both techniques are equally useful when the restrictions are met. So, for example, when integration is possible and when the intervals of the curve being looked at are "small enough" such that the line connecting the limits of the interval is a straight line, respectively.

Conclusion:

The only uncertainty in this lab was the inaccuracy of the method of using the average to find the area under the curve manually. However, with smaller intervals the uncertainty becomes smaller, and with "small enough" intervals the uncertainty becomes so small that it is considered negligible.

As was explained previously, the value found by solving the problem numerically was very close to the value of solving the problem analytically. With significant figures, there would be no difference between the two and they would be considered equal.

It can be known when the interval chosen is "small enough" by comparing the numerical value to the analytical value. If the analytical value was not known such that it cannot be compared with the numerical value, it can still be known when the interval chosen is small enough by looking at the change in position. If it is so small or so close to zero such that it can be neglected, as in our problem, then we know that we are close enough and the interval is small enough.  

2 March 2015: Free Fall Lab- Determining the value of gravity using a sparker

Purpose:

The purpose of this experiment was to graphically calculate the value of the acceleration of gravity using data gathered from a sparker, an apparatus which labels the position of a freely falling object every 1/60th of a second on a piece of tape. 

Apparatus:

                                    Picture 1: A photograph of a sparker

A sparker was used in this experiment to gather position data required for calculation of the value of gravity. The sparker consists of the metal column, a cylinder with a metal ring that acts as the freely falling object, sparker tape that runs across the inner side of the column, and a sparker generator that generates 60 Hz of current. The sparker works by holding the cylinder at the top of the metal column by electromagnetic forces. When the electromagnet is turned off, the sparker generator creates a spark pulse every 1/60th of a second that is absorbed and released by the metal ring around the cylinder, generating a dot on the sparker tape. This is how the motion of the freely falling wooden cylinder is recorded and monitored.The sparker tape that was used in the experiment is seen in the picture below:

    Picture 2: Some sparker tape used in the free fall experiment. The dots can be seen on the closer     end of the tape

The sparker tape used in this experiment is about 1.5 m tall, giving a large amount of data. Eighteen of the many dots were chosen to be used as the position data. Some of the chosen dots can be seen in the picture below. As can be seen, the spaces between the dots increase as the tape goes along. This is due to the velocity of the  freely falling cylinder increasing due to the acceleration of gravity.

    Picture 3: A closer view of the sparker tape used in the free fall experiment

Abstract:

The position of the freely falling cylinder can be measured by setting one of the dots on the sparker tape to be the origin (at time 0 right before the fall) and measuring the distances the dots are from the "origin" dot. Knowing that the distance between each consecutive dot is the distance the cylinder traveled in 1/60th of a second, the average velocity between two dots can be calculated by dividing the distance traveled between the two dots by the time (1/60th of a second). Using the position and velocity data, a graph of position vs. time and velocity vs. time can be constructed, and the acceleration of gravity can be found. The value of gravity can be calculated from the velocity graph by looking at the slope or by taking the derivative of the velocity function. The velocity function is in the form y = mx + b, which is comparable to the kinematic equation V = Vo + at, where a is equal to m and b is equal to Vo. It can also be calculated by taking the second derivative of the position function, which is in the form y = Ax² + Bx + C. This is comparable to the kinematic equation  X = Xo + Vo(t) + (1/2)at², where A is equal to (1/2)a. 

Procedure:

 Once the sparker free fall experiment was performed and the distance between the dots was measured, the data from the tape was recorded into an Excel spreadsheet, with time (in seconds) in one column and distance (in cm) in another. The distance was measured as the distance from each dot to the "origin" dot, and the time was measured proportionally in 1/60th of a second for every dot after the "origin" dot. Delta x was then found by finding the distance between each of the two consecutive dots. The mid-interval time was then recorded; it is the time where the position of the cylinder was right in between the two dots (initial time + 1/120). Then, mid interval speed was calculated by dividing delta x by 1/60. The mid interval speed can be used to find average velocity because acceleration is constant. The area of a rectangle located at half the interval is equal to the area under the curve when acceleration is constant. Then, mid interval speed and time were used to construct the velocity graph seen below, with a linear trendline.

    Table 1: Data gathered and calculated from the sparker tape

    Graph 1: Velocity vs time graph of the freely falling cylinder in the sparker

The distance and time columns were then used to construct the distance vs time graph below, with a polynomial trendline of order 2.

    Graph 2: Distance vs time graph of the freely falling sparker object, where the origin is at the top of the sparker

As stated above, the acceleration can be found from the slope of the velocity graph or from multiplying A by 2 in the distance graph. However, the acceleration of gravity in this experiment was calculated from the velocity graph. The value of g was calculated to be 941.5 cm/s², or about 9.42 m/s², a percent difference of about -3.98% from the real value of g (9.81 m/s²). Percent difference can be found by subtracting the experimental value with the real value, then dividing by the real value and multiplying by 100%. 

In order to minimize error and uncertainty in this experiment, the class' values of g were pooled and the standard deviation was found. The standard deviation shows the variation from the mean. The average of these values was found to be 957.7 cm/s², or around 9.58 m/s². The standard deviation of the average was then found by subtracting each experimental value of g by the average value of g (called the deviation), then squaring each deviation and taking the average of all of them. The standard deviation was then found by dividing the average squared deviation by the number of "trials" (9 in this case) then taking the square root. The final experimental value of g was found to be 957.7 +/- 18.1 cm/s². The table of these values is found below:

    Table 2: Class' pooled values of g, average g, and standard deviation of the average of g

Conclusion:

As can be seen, the real value of g is more than one standard deviation away from the experimental value of g. Therefore, it can be concluded that the sparker is not a reliable apparatus for measuring and calculating the value of g. Two major assumptions done in this experiment that could have greatly affected the results are the neglecting of friction and other outside forces other than gravity as well as the sparks being exactly 1/60th of a second apart, with the first assumption having a much greater effect. The pattern seen in most of the trial values of g and the average value of g is that it is smaller than the real value of g, which is good proof that gravity is not the only force acting on the freely falling object but also other forces such as friction that combat the force of gravity. This is a systematic error, or an error that results in having results that are either too low or too high, but not both.  The assumption that the sparks are 1/60th of a second apart does not affect the results of the experiment by much, if at all, since the spark frequency is quite accurate.

Another assumption made that has some effect on the results of the experiment is that the sparker was dropped the same way each time. However, in reality the sparker does not drop the freely falling object in the same way each time, resulting in more or less outside forces on the object each time. This can be seen as random error, or error that results in results that are both too high and too low.

The point of the part of the lab after finding the experimental value was to minimize any sources of error or uncertainty by pooling the class' data and calculating for the average value of g. It was also done to find the variation and spread of the class data through the calculation of the standard deviation. The standard deviation of the mean was also calculated to show the accuracy and precision of the apparatus used, which was found to not be very accurate. It was also used to find the level of confidence in the mean of the data. The level of confidence is another way of showing the accuracy and precision of the apparatus.

(Note to instructor: The rest of the required questions in the handout are answered throughout the post.)

23 Feb 2015: Finding the equation that relates period and mass of an inertial pendulum

Purpose:

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.