(figure showing the set up with two small wooden blocks held in clamp)
A piece of paper with a vertical mark on it was placed behind the pendulum so that when the latter was at rest it hid the vertical mark from an observer standing in front of the pendulum.
The length L of the cotton thread was measured from the point of attachment to the center of mass of the bob by meter rule. (This was called effective length of the pendulum.)
The pendulum bob was set to swing through an arc of about 10
The time for 20 complete oscillations was measured with a stop-watch by setting the stop-watch goes when the pendulum was released from a little angle apart from the vertical.
Successive amount of about 10cm was shortened from the length of the pendulum by putting the cotton thread through the vice and for each new length we took observation of the time for 20 oscillations.
The experiment with different length was repeated .
A graph with values of T2 / s2 as ordinates against corresponding values of L/ m2 as abscissa should be plotted to yield a straight line.
By applying,
T == 2π
the acceleration due to gravity can be found from the slope of the graph.
Data and Data Analysis
By using vernier caliper, the length from the hook of the bob to its center of mass was 2.73cm.
The effective length was found by adding the length of string and radius and the
length of hook of the bob.
Result table:
Result graph showing the relation between the period and effective length can be used to find the acceleration of free fall
The slope of the graph =
= 4.15ms-2
g =
= 9.52 ms-2
Result with error and accuracy
The graph is not precise enough. The best fit line may have deviation with the original plotted graph. It causes the error in calculating the slope and so the acceleration of the motion.
Absolute error of length = 0.2mm
Absolute error of time = 0.01s
Experiment value of g = 9.52 ms-2
g =
Find the for each experiment by using the above formula:
Average value of :
=
= 0.114
The value of g with precision is ( 9.52 0.11 ) ms-2
Compare:
The standard value of g = 9.81 ms-2
The values gained from the graph g’ = 9.52 ms-2
The percentage error = X 100
= X 100
= X 100
= 2.96
The experiment result of is 2.96 smaller than the expected value 9.81 ms-2
It is quite accurate as the experiment value does not deviate a lot from the expected value.
Discussion
- Random Error
There are some limitations in the experiment due to instrumental limit (reading error) which cannot be reduced by repeating the experiment.
- Percentage error due to scale uncertainty
The percentage error due to the limitation of measuring length:
x 100
Find the percentage error of each measurement by using the above formula:
Average value of the percentage error due to this limitation:
=
= 0.198
The percentage error of measuring time:
x 100
Find the percentage error of each measurement by using the above formula:
Average value of the percentage error due to this limitation:
=
= 0.0246
Both of the percentage errors are small and they are acceptable. And the percentage of measuring length is larger than that of measuring time. It is more significant. However, it is not the major source of error.
- Percentage error due to external factors
This kind of errors results from unknown and unpredicted variations in experimental situations. They cause some fluctuations in the values of repeated measurement.
They may due to:
- reaction time when pressing the button of the stop-watch
- occurrence of elliptical oscillation or called circular pendulum motion when the string is long or the bob is not released on the same plane of its motion
- occurrence of the self-rotation of the bob (it is also called torsional oscillation) which may cause the energy lost and influence in the period of motion
- existence of air resistance which may cause damping and influence on the period of motion
- unbiased estimates of measurement readings by the observer
The above random errors are small and can be reduced by repeating the experiment.
- Systematic Error
Systematic error causes all the measurements to be shifted systematically in one direction – either larger or smaller than it would be. They cannot be reduced by repeating the experiment.
They may due to:
- parallax in reading the scale when viewing the scale always from one side
- a zero error on the ruler or stop-watch
- the pendulum released from a large angle
- extensible string affecting the movement of the pendulum
Improvement:
- The parallax error can be prevented by viewing the scale perpendicularly.
- Always check and adjust the zero reading before using an instrument. If nor, add or subtract the zero error from all of the readings.
- Release the pendulum from a smaller angle.
- Use an inextensible string in the experiment.
- Implications, suggestions and improvements
(Referring to the questions of reflection stated in lab menu)
- Errors in timing occur both when the stop-watch is started and when it is stopped (reaction time). These errors are unlikely to be less than the interval at which the second hand moves (scale uncertainty). What type of error they are (random or systematic)? How are they related to the final error in the experiment?
Ans: The reaction time of human is about 0.02 s. As starting and stopping the stop-watch involve the time delay in both aspects, the total error accumulated is 0.04s. The significant figure of measuring time is 0.01s. It is smaller than the error caused by human reaction. It is a kind of random error. Considering the final result of the experiment, this error will lead to more time measured. The larger the value of T, the smaller the value of g. Random error can be reduced by repeating the experiment. To compensate this error, it is suggested to use a more accurate instrument in measuring the time, for example, using computer data-logging system.
- The error in the experiment of L is the error inherent in the use of any scale (half the distance between adjacent markings, doubled because of two ends to the distance measured). What type of error they are (random or systematic)? How are they related to the final error in the experiment?
Ans: There is inherent error in measuring the length. The values depend on the observer’s estimation when the value lies within the deviation of the instrument. The error includes taking reading from two ends of the ruler. It is a kind of reading error caused by scale uncertainty.
-
As the value for g is obtained solely from the slope of the graph it follows that the % error in g is the same as the % error in the slope. Estimate the difference between the slope of your chosen ‘best’ straight line through the points and the slope of other possible straight lines drawn through the points and express this in percentage. State your value for g accordingly.
Ans:
(The mean value obtained by the graph plotted by hand is different from that obtained from the graph done by software. There may have deviation of values.)
Percentage error in g in the slope:
= x 100
=3.86
Using the value of centroid and applying the corresponding formula, the value of g calculated = 9.79 ms-2
The straight line with slope m is the best fit line plotted.
Percentage difference between the slope m and m+
= x 100
= x 100
= 14.73
Value of m+ 14.73 % larger than that of m
The value of g = 9.79 x (1+14.73%)
=11.24 ms-2
Percentage difference between the slope m and m-
= x 100
= x 100
= 13.77
Value of m 13.77 % larger than that of m-
The value of g = 9.79 x (1-13.77%)
=8.45 ms-2
- Suggest a method to measure the period of oscillations other than that used in the experiment. Comment on the two methods.
Ans: Computer/multimedia system can also be used in this experiment to measure the motion of a simple pendulum as it passes through a photogate timer. The computer can plot a graph of the results.
The photogate emits a beam of infrared light from one side to which is received by a sensor to provide the timing signal. With the software, data is collected by recording the times when the infrared light beam between the jaws of photogate is interrupted. To time a pendulum using a photogate and LoggerPro, the pendulum is allowed to swing between the jaws of the photogate. The period is then plotted as a function of time on LoggerPro.
In pendulum mode, the timer measures the period of one oscillation. Timing begins as the pendulum first cut through the beam. The timer ignores the next interruption, which corresponds to the pendulum swinging back to the opposite direction. Timing stops at the beginning of the third interruption, as the pendulum complete one full oscillation.
Using this method to carry out the experiment can prevent some human error caused by the original. It would not consist of the error of reaction time. And also, the data can be obtained by the function of software. It prevents lots of calculating error of human. The valued obtained is much accurate than the original one.
- If the experiment is performed with the pendulum suspended from an inaccessible point, e.g. the ceiling (i.e. you cannot measure the length of the pendulum), suggest a modification to the experiment to find the value of g.
Ans: It is an alternative method to find the acceleration due to gravity with the help of ticker-tape timer
(The length of the string is not necessarily known.)
The separation (x) between the dots printed by the ticker-tape timer, the time for each of the position can help find the instantaneous velocity of the corresponding time by working out the slope of x-t graph plotted. Plot a suitable v-t graph. Again at suitable points on the v-t graph, find the acceleration by working out the slopes. Period of the oscillation can be found.
The acceleration due to gravity is found.
Conclusion
By means of a simple pendulum, the acceleration of free fall was found to be ( 9.52 0.11 ) ms-2 which is 2.96% smaller than the expected value 9.81 ms-2