L10: Period of a Pendulum
Author: Heide Carrion Lab Partner: Bryan Atkinson
PHY 1048L, SPC Gibbs Campus Dr. Ann Marie Eligon Monday, 11 November 2011, 2:00 pm
Carrion 1 Introduction
The purpose of this lab was to find out which parameters affect the period, , of a
pendulum. It was also to experimentally find a value for the acceleration due to gravity, , by
measuring the period, .
Theory
Period of a Pendulum
Where
the length of the
pendulum from pivot point to center of mass and
is the
acceleration due to gravity.
The actual period, , is a complex function of angle,
, but for small angles that are less
than or equal to 35 , the above equation can be used to calculate the period. The data gathered in this lab will show that for small angles the period, T, is the above function of length, L and depends upon length and acceleration due to gravity, g. Natural Log of Period of a Pendulum
When plotting T vs. L data it is difficult to see the relationship between the curved p lot and the equation for period of a pendulum, T. Taking the natural log of the equation results in the above equation for natural log of the period, ln T. Plotting the data for ln T vs. ln L results in a straight line graph which more readily demonstrat es the relationship between T and L.
Carrion 2 Equation for a
Where:
Straight Line
In the linear plot for the natural log data, it is easy to see if the equation for the period, T of a pendulum as a function of length, L, is true by how close to a straight line the points fall on the graph and how close b is to 0.5. Equation for the Acceleration Due to Gravity as calculated from the T vs. L experimental
data Using the data from the straight line graph, we can also see how close the value for g
comes to the accepted value for acceleration due to gravity of 9.80 . Using this method is the same as setting g equal to 9.8, and then comparing the value for a to the intercept. Equation for the Acceleration Due to Gravity by direct measurement of T and L.
When L and T are measured directly for a large pendulum, the results are more precise and the calculation for g will therefore come very close to the accepted value for g by using the above equation.
Carrion 3 Procedure P art
I: First we measured the masses of two different bobs, one larger than the other, on a zeroed
e-balance. Next we tied a length of string around the smaller bob and set up a pendulum apparatus, making sure that the string was as long as possible. We measured it from pivot point to the center of mass for the bob. Then we started the pendu lum swinging and timed 20 cycles with a stopwatch using the same reference point at the bottom of each cycle. We used the data to calculate the period of the pendulum. This process was repeated for the larger bob, keeping the length of string constant from the pivot point to the center of mass for the larger bob as it was for the smaller bob. P art
II: Using the last pendulum setup, we set the arc of the pendulum to fifteen degrees using a
protractor. Next we timed 20 cycles and used the data to calculate the period. This process was repeated for an arc of thirty five degrees. P art
III: In this part we needed to use six different string lengths ranging from 0.3m to 1.0m. We
had to find the longest measurement because 1.0m would have been too long for our setup. After we found the closest measurement possible to the upper limit, which turned out to be 0.892m, we found that the lower limit would be 0.300m. We calculated six data points which uniformly covered this range. Still using the larger bob, we timed and recorded data for 20 cycles for each length, starting with the longest length. We kept the arc at approximately twenty five degrees for
Carrion 4 the first four lengths and then reduced to ten degrees for the two shortest lengths in order to stop the short pendulums from swinging chaotically. P art
IV: Finally, we used the data from a large pendulum to calculate a value for acceleration due
to gravity, g . Using a tape measure, we measured the length from the pivot point to the center of mass for the bob. Then the photogate timer was used to obtain the period of this pendulum. Each group made their own measurements of the large pendulum. All data for length, L and period, T from each group was recorded in order to be averaged and used to make the final g calculation.
Results P art
I: Mass of Pendulum
Period of Pendulum
0.016kg
1.771s
0.086kg
1.772s
These results verified that within experimental error, the period, T, of a pendulum is independent of mass as predicted. P art
II: Arc or Angle of Pendulum
Period of Pendulum
15
1.774s
35
1.814s
Carrion 5 These results verified that the period of a pendulum is not dependent upon the angle if it is relatively small as predicted. In comparing these values to the period values from Part I, one can see only a small change. This is reasonable after taking experimental error into consideration. Also, after taking into consideration that the lengths of all four pendulums in Part I and Part II were the same, one can still predict that the period of a pendulum will indeed be dependent upon the length of the pendulum. P art
III: Slope, b = 0.5241 Experimental value of g = 10.28
These results verify that the period of a pendulum is a function of the length of the pendulum and is also depende nt upon the acceleration due to gravity as predicted. After plotting the natural log of the dat a points for both length and period, we obtained a straight line. These results also indicate that within reasonable experimental error, o ur value for slope, b, comes very close to 0.5 and our experimental value of g is reasonably close to the accepted value for acceleration due to gravity. P art
IV: Length of Large Pendulum: 2.40 ±1.16E-02 (4.84E-03%)m Period of Large Pendulum 3.112 ±3.068-03 (9.859E-04%)m Calculated value of g : 9.774 ±6.662E-04 (6.816E-03)
These results indicate that the calculated value of g is extremely close to the accepted value of g. After considering the percent error, the calculated value of g cannot be closer to the actual value of g than 9.8. After analyzing the resulting data, it is clear that there was a larger
Carrion 6 error in the measurement of length of the pendulum which resulted in the erro r for the calculation of g. Error Analysis The first random error is the aforementioned error in length o f the large pendulum. This error is probably due to the fact that all groups took a measurement of the length. The technique of each person taking the measurement may have varied. Our group used a measuring tape because it seemed easier than t rying to balance a meter stick while standing on a chair. The measurement was taken in centimeters and then converted to meters. Others may have use the meter stick or another tool, perhaps each with a different least count, or they may have taken the measurement in millimeters and then converted to meters per the lab handout instructions. The measurements were close enough but not all measurements were precise. Our group¶s measurement was the most different. This is why measurements were taken by each group and then averaged in order to get a mean value for the length of the pendulum. A second random error was the start and stop times of the measurement of the cycles. We used a stopwatch which was manually started and stopped. Also, the same person always started, stopped and read the watch, while the same person a lways started the pendulum swinging and counted the cycles. While this gave consistent results, there is also the margin of human error. Each person has a different technique and reads the scale the sa me way each time. It is impossible to be completely precise when working with these factors. A third random error is the least count of the equipment. The e-balance is precise only to 0.001kg. The stopwatch is precise only to the 0.01s. The meter stick is precise to only 0.001m.
Carrion 7 Using these tools limits the measurements and therefore limits the calculations and resu lts, making it impossible to know true dimensions. One systematic error is the hanger used to secure the pendulum. It caused the string to hang slightly askew which may have caused an error when measuring the angle at 35 degrees. This could have caused the angle to be slightly more than 35 degrees, in which case the period becomes angle-dependent and the above equation for the period would not suffice. This may be evident in the value obtained for period for the fourth pendulum from Parts I and II. While the value is reasonably within experimental error, it is markedly different from the first three values. A second systematic error for this experiment is that air resistance was not taken into consideration. Air resistance causes the pendulum to decelerate over a span of t ime, therefore slightly increasing the time of each cycle. These increases in period, while miniscule, add up and affect the results. While timing a maximum of 20 cycles for the pendulum helps reduce this error a bit, a correction factor would also have to be introduced. In an ideal situation, this experiment would have been performed in a vacuum or at least in a more sophisticated setting where the air resistance could be accurately measured and used in the calculation of the periods. A third systematic error was found in the change of the string lengths. The first four string lengths were the longest. These worked well with a starting angle of approximately 25 degrees. We were able to time 20 cycles with no problems. However, the starting angle of the two shortest lengths had to be reduced by approximately 15 degrees. We found that any angle larger than approximately 10 degrees would not work for the shorter lengths. When set to swing at the larger angle, the swing o f the shorter lengths became chaotic much before t he completion of 20 cycles by changing direction, rotating, and crashing into the support stand. After reducing
Carrion 8 the angle we were able to time and complete the requisite 20 cycles. This introduced an error of inconsistency. It is easy to see why measuring the length and time of the larger pendulum is more accurate and results in a better value for g. Conclusion In this lab we learned that the period of a pendulum does not depend on its mass. Also for angles of 35 degrees or less, the period of a pendulum only depends on the length and the acceleration due to gravity, g . We also determined, through the experiment, that the value of g can be calculated by graphing the natural log data for the period, T, as a function of length, L. We were also able to find a value for g by directly measuring the period and length of a pendulum. In this way, we can find errors in data by co mparing how close our calculated value of g comes to the accepted value for acceleration due to gravity, g.
