1. Introduction

2. Setup and collected data

3. Nonlinear method

4. Linear method

5. Least squares method

6. Discussion and conclusion

7. Links

8. References

The data was collected on March 2, 2020, during a lab exercise regarding the subject PHYS114. The data is to be graphically presented using three different methods: nonlinear method, linearization and least squares method. In addition, calculated values for \(g\) and \(k\) is to be compared with theoretical values.

In the experiment, the acceleration of gravity and the pendulum´s inertia of radius, will be determined with different methodes which looks at time period for different pendulums lengths. The relationship between these sizes is given by

The equipment used in the experiment are listed in table 1.

The counter, oscillioscope and DC voltage source were connected to the pendulum mounting consisting of a knife egde and a photocell. The voltage difference was set to 5V. The oscillioscope was set so that an entire period of signal showed as the pendulum swung. The wingnuts on the base plate of the mounting were sligthly adjusted to make sure the base plate was stable. It was ensured that the pendulum hung centered on the mounting and that the pendulum swung symmetrically about the ligth beam from the photocell. The figure below shows a drawing of the pendulum mounting.

The mass center for the pendulum was found by balancing it on the top of the knife edge. When the point of balance was found, two lines were drawn with a pen, one on each side of the knife edge. The pendulum used had ten different holes. By hanging the pendulum on the knife edge through the various holes, ten different pendulum lengths were used. The figure below shows an illustration of the pendulum, with its holes and the lines marking the mass center.

The caliper and metal knobs were used to do combined measurements of the lengths from the holes to the mass center. The distance from hole 5 to the center of mass was measured first, and this distanse was measured ten times - five times to the nearest line and five times to the line furthest away. The mean value and standard deviation were found by the two following equations

The result is shown in table 2.

Each of the remaining holes were measured once and given the same uncertainty as for hole 5. Holes 1-5 were given negative values, while holes 6-10 were given positive values. The mass center was located between these two groups. The results of the lengths are inserted in table 3. The value of \(l_5\) in the table is the mean value from table 2.

The total length \(L\), width \(W\) and heigth \(H\) of the rod were measured with the caliper, and the result is to be found in table 4.

The values were used to calculate the theoretical value for the inertial radius k, given by

With this formula, theoretical value of k was calculated to be \(k=106,35\)mm.

With the knife edge through hole 1, the time period T was measured ten times starting from an small angle. The standard deviation and mean value of T were found by the following two equations

Using the equations given by

the maximum angular \(θ_0\) was determined by the uncertainty, or the standard deviation of T, and the mean value. The results for the time period for small angles, with related mean value and standard deviation and maxium angular are all inserted i table 5.

Using the angle \(θ_0\) as starting position for the pendulum, the time period was measured once for all ten pendulum lengths. All time periods were given the same uncertainty as for hole 1. The resulta of the times are given in table 3.

The magnitude of the acceleration of gravity varies from place to place on earth. At 60 latitudes the acceleration is \(g_{60°}=9.8192m/s^2\), and at 65 latitudes it is \(g_{65°}=9.8229m/s^2\). The experiment was executed in Bergen, which is just over 60 latitudes, and therefore the acceleration in the laboratory should be between \(g_{60°}\) and \(g_{65°}\) [1].

The results of the measurments listet in table 3 were used to plot T as a function of l. Two seperated plots for negative and positive lengths were also created, including uncertainties. A regression line was drawn for these two plots (N=4). Based on the graphs, a value T´was selected and plotted parallel to the x-axis. This line crossed each of the regression lines two times, thus four times in total. The crossing points were called \(l_2\)´and \(l_1\)´for the negative lengths, and \(l_1\) and \(l_2\) for the positive lengths, from left to right on the x-axis. The values of the points were found. The uncertanities are given by

where

The mean value of \(l_1\) is given by

and the mean value of \(l_2\) is given by

The uncertainties of the mean values are given by

The acceleration of gravity, g, is given by the following equation

and the uncertainty is given by

The pendulum´s inertia radius, k, is determined by using the following equation

and the related uncertainty is given by

The figure below shows a plot of \(T\) as a function of \(l\), with the values from table 3. The negative values og the lengths are represented with blue squares, and the green dots represents the positive values. The code used to plot is nonlinearmethod.C

Two seperated plots for the negative and positive lengths are shown under, with uncertainties. These are made with the codes negativelengths.C and positivelengths.C. Fit Panel in ROOT was used to make a smooth line between the datapoints. The previous plot, with alle the values, shows that a line of \(T´=0,95\) would cross a regression line for the points fire times. The line \(T´=0,95\) were therefore included in the plot for the negative and positive lengths.

By visual estimate, \(l_2´\), \(l_1´\), \(l_1\) and \(l_2\) was determined to be

Equation (8) gives an uncertainty of

The mean value of \(l_1\) and \(l_2\) were found by equation (9) and (10) to be

Using equation (11), the uncertainities of the mean values are

Thq equations (12), (13), (14) and (15) were used to calculate g og k. The results are

Observe that both of the values are lower than the theoretical ones.

*I failed to perform this task as planned. I wanted to create a grade 4 polynomial to fit the datapoints, but unfortunately I was unable to do that. Then I could have used the function and \(T´\) to get more accurate values of \(l_1´\), \(l_2´\), \(l_1\) and \(l_2\). Which may have resultet in better results of \(g\) and \(k\). In stead I used Fit Panel in ROOT to make a smooth line between the points and then used visual estimate to determine the crossing points. It was difficult to see exactly where the crossing points were, which probably leads to a considerable uncertainty in the determination*

From equations (1), we can get the expression

It is on the linear form y=ax+b, where \(y=l^2\), the slope is \(a=g/(4π)^2\), \(x=lT^2\) and the constant term is \(b=-k^2\). If a is known, the acceleration of gravity is given by

and the uncertainty is given by

When the value of b is known, the pendulum´s interia can be found by

and it has an uncertainty given by

Based on the measured lengths, where all are treated as positive, and the measured time periods T, \(x=lT^2\) and \(y=l^2\) were calculated along with their uncertainties. The uncertinties is to be found using the following equations

The values were used to plot \(y=l^2\) as a function of \(x=lT^2\). Then Fit Panel in ROOT were used on the same plot to make a linear fit of the datapoints. The slope and constant were found and used to calculate \(g\) and \(k\).

Table 6 shows the calculated values og \(x=lT^2\) and \(y=l^2\) with uncertainties. The uncertainties are derived from equation (21) og (22).

The figure below shows \(l^2\) as a function of \(lT^2\). The plot was made in ROOT with the code
linearmethod.C

The linearization made in Fit Panel is shown in the figure under

The linear function has a slope of value

and the value of the constant term is

By the equations (17) and (18), the acceleration of gravity was calculated to be

The pendulum´s intertia was calculated by the equations (19) and (20) to be

The calculated values for both g and k are lower than the theoretical ones, even with the uncertainties.

A table of data points and uncertainties was created for the linear chart, with a column for \(s_i´\), derived from the equation

The slope \(â_1\)and the constant term \(b̂_1\) were found by the two following equations

The two associated uncertainties are given

With this estimate of \(â_1\) and \(b̂_1\), \(g\) and \(k\) were found by equation (17) and (19),respectively. The uncertainty of \(g\) was found by equation (18), while the uncertainty of \(k\) was found by the equation (20). To find the sum of squares \(SSE_1\), the estimated values \(â_1\) and \(b̂_1\) were used together with the following equation

In this equation, the term \(s_{l^2}\), which takes into account the uncertainty along the y-axis, were used. \(s_i´\) also takes into account the uncertainty along the x-axis. Corrected values for the slope and constant term, \(â_2\) and \(b̂_2\), was determined using \(s_i´\), as given in the following equations

The uncertainties of the corrected \(â_2\) and \(b̂_2\) is given by

There corrected values was then used to find new values of \(g\) and \(k\), as well as a corrected value for the sum of squares, \(SSE_2\), given by

Table 7 shows the data points with uncertainties for the linear chart and \(s_i´\) calculated with equation (23).

With equation (24), (25), (26) and (27), the slope \(â_1\) and the constant term \(b̂_1\) was found to be

With these values, as well as the equations (17), (18), (19) and (20), \(g\) and \(k\) were calculated to be

The sum of the \(SSE_1\)-values, calculated using equation (28), is also inserted in table 7. By using equations (29), (30), (31) og (32), corrected values of the slope \(â_1\) and the constant term \(b̂_1\) was found to be

New values for g and k was then calculated to be

The correected calues of squares, \(SSE_2\), found by equation (33), is also to be found in table 7.

The sum of the corrected square sum is considerably smaller than when only the uncertainties along the y-axis were taken into account. This means that \(â_2\) and \(b̂_2\) form a curve that fits the measurement points better than \(â_1\) and \(b̂_1\) do. Both before and after the iteration, both the values of g found with this method, is lower than both \(g_{60°}\) and \(g_{65°}\). The actual value lies between the acceleration at \(60°\) and \(65°\) latitude. With uncertainties, \(g\) hits within this range, although the calculated values are slightly low. The calculated value of \(k\) with uncertainties is not within the theoretical \(k\). Here, too, calculated value is lower than theoretical.

Three different methods were used to determine the acceleration of gravity, \(g\), and the radius of inertia, \(k\), for a physical pendulum. This was done by looking at how the time period depends on the length of the pendulum.

The mass center of the pendulum was found by balancing it on the knife edge. A point where the rod was absolutely at rest was not found, but it was almost at rest. Therefore, the correct mass center may not have been found, but since the pendulum was almost at rest, it can not be far away. When the lines were drawn, the pendulum was tilted up to each of the sides, and it is possible that the pendulum was slightly shifted in this movement. Positive for the non-linear method is that any systematic errors in the determination of the mass center will be offset, because mean values of \(l_1\) and \(l_2\) are used. This does not apply to the linearization. A disadvantage of the linear method occurs if the mass center is determined incorrectly. Then the points with the same calculated distances to the mass center, only on different sides of the mass center, will have different distances to the actual mass center. This means that the values of \(l\) on one side of the mass center will be larger than they should be, and the values of \(l\) on the other side will be smaller than they should be. The plot of \( l ^ 2\) as a function \(lT ^2\) will then give an incorrect representation of the system where the points deviates from a straight line. Since the points in the linear representations above are approximately on a straight line, this indicates that the mass center is determined with sufficient accuracy.

The ten measurements of the time period with the knife edge through hole 1 of the pendulum were used to determine the maximum angular, \(θ_0\), that the pendulum could have. Ten more attempts, one for each hole, were then made starting from this angle. But there is a considerable uncertainty as to whether the pendulum was actually released from this angle. If the pendulum swung with an angle too large during the entire measurement series, the value of \(g\) and \(k\) would both be too small.

In Table 8, the values of \(g\) and \(k\) found by nonlinear method, linear method and least squares method, are listed. The SEE-values found by the least squares method are also included.

The theoretical value for the acceleration of gravity in the laboratory in Bergen lies between \(g_{60°}=9.8192m/s^2\) and \(g_{65°}=9.8229m/s^2\). The values calculated by the least squares method hits within the interval when the uncertainties are included. Since both values are small, it is the value found first that predicts \(g\) the best, because it has the greatest value of the two. The uncertainty is also less. But the difference is small. The value of \(g\) found by the nonlinear method is lower than the theoretical one, even with uncertainties. The non-linear method gave the lowest value and the worst result. This may by because of the great uncertainty associated with the visual estimate of the crossing points. Thus, the first use of the least squares method is considered best of determining \(g\) in this case. The figure below illustrates this result, the plot is made by the code gvalues.C. The horizontal line represents theoretically \(g_{65°}=9.8229m/s^2\). 1 on the x-axis shows the value of the nonlinear method, 2 the linear method and 3 the first and 4 the second use of the least squares sum metod.

The theoretical value of \(k\) determined by formula (4) has a value of \(k=106,35 mm\). The value was found based on measurements of the pendulum's width and length. The measurements have some uncertainty, but this is not taken into account. All methods resulted in calculated values of \(k\) which was lower than the theoretical one. Llinear method and the second use of least squares method, with uncertainties, came closest to the theoretical value. Also for \(k\), the nonlinear method gave the worst result. The figure below, made by the code kvalues.C, illustrates the results. The horizontal line represents theoretically \(k\). The x-axis refers to the different methods in the same way as for the figure of the calculted \(g\) values.

- Codes (the codes that were used to plot in ROOT)
- Plots (pictures of the plots made by the codes)
- Eqations (pictures of all the eqations listed in this website
- Tables (pictures of all the tables in this website