Using the measuring tape; measure out 90cm on the 2m string then using the scissors cut the string, while making sure that the string is cut above 90cm as some measurements are required for the purpose of tying.

Tie one end of the strong string onto the center of the pendulum and the other end of the string onto the pivot point (end of the whiteboard diagram hanger), while making sure that the actual length from the pivot point to the center of the pendulum is 90cm.

Using the whiteboard marker and ruler, rule a line directly from the pivot point to the center of the pendulum (NOTE: the length of the line should be 90cm) on the whiteboard.

Rule a 7.9cm line horizontal to the left from the point where the center of the pendulum is located.

Using the ruler and whiteboard marker, join the pivot point to the end of the 5cm horizontal line.

Slowing lift the pendulum to the recent ruled line, while making sure that the string connecting to the pendulum and pivot point remains straight.

Release the pendulum slowly.

Allow the pendulum to swing for two cycles then using the stopwatch start the timer.

Stop the timer when the pendulum reaches ten cycles, excluding the first two cycles. This gives the it time it took to cycle 10 times.

Repeat steps 8-11 three more times.

Repeat steps 3-12 but using a 60cm string with a 5.2cm horizontal line to the left from the point where the center of the pendulum is located and 30cm string with a 2.6cm horizontal line to the left from the point where the center of the pendulum is located.

Results

The Time it Took for a Pendulum to Swing and Cycle Ten Times

Resolution Ruler – 0.1cm Stop Watch – 0.01s

Calculations

Calculating the gravitational acceleration

T = 2π

T = 2π

g =

Calculating Gravitational Acceleration for 0.30m

10.8s per 10 pendulum swing cycle = 1.08s per pendulum swing cycle

L = 0.30m and T = 1.08s

g =

g = 10.2ms-2

Calculating Gravitational Acceleration for 0.60m

15.7s per 10 pendulum swing cycle = 1.57s per pendulum swing cycle

L = 0.60m and T = 1.08s

g =

g = 9.6ms-2

Calculating Gravitational Acceleration for 0.90m

19.0s per 10 pendulum swing cycle = 1.90s per pendulum swing cycle

L = 0.90m and T = 1.90s

g =

g = 9.8ms-2

Calculating Uncertainties for the gravitational acceleration

0.30m Pendulum

Since T = 10.8 and L = 0.30, the uncertainty for T = 10.8s ± 0.05s and L = 0.30m ± 0.05m

Highest value for the gravitation acceleration using 0.30m pendulum is;

L = 0.30m + 0.05m

= 0.35m

T = 10.8s – 0.05

=10.75s per 10 cycles

g =

where L = 0.35 and T = 1.075s per cycle

g =

g = 11.9ms-2

Lowest value for the gravitation acceleration using 0.30m pendulum is;

L = 0.30m - 0.05m

= 0.25m

T = 10.8s + 0.05

=10.85s per 10 cycles

g =

where L = 0.25 and T = 1.085s per cycle

g =

g = 8.4ms-2

0.60m Pendulum

Since T = 15.7 and L = 0.60, the uncertainty for T = 15.7s ± 0.05s and L = 0.6m ± 0.05m

Highest value for the gravitation acceleration using 0.60m pendulum is;

L = 0.60m + 0.05m

= 0.65m

T = 15.7s – 0.05

=15.65s per 10 cycles

g =

where L = 0.65 and T = 1.565s per cycle

g =

g = 10.5ms-2

Lowest value for the gravitation acceleration using 0.60m pendulum is;

L = 0.60m - 0.05m

= 0.55m

T = 15.7s + 0.05

=15.75s per 10 cycles

g =

where L = 0.25 and T = 1.575s per cycle

g =

g = 8.8ms-2

0.90m Pendulum

Since T = 19.0 and L = 0.9, the uncertainty for T = 19.0s ± 0.05s and L = 0.90m ± 0.05m

Highest value for the gravitation acceleration using 0.90m pendulum is;

L = 0.90m + 0.05m

= 0.95m

T = 19.0s – 0.05

=18.95s per 10 cycles

g =

where L = 0.95 and T = 1.895s per cycle

g =

g = 10.4ms-2

Lowest value for the gravitation acceleration using 0.90m pendulum is;

L = 0.90m - 0.05m

= 0.85m

T = 19.0s + 0.05

=19.05s per 10 cycles

g =

where L = 0.85 and T = 1.905s per cycle

g =

g = 8.2ms-2

Discussion

Theoretically the acceleration due to gravitation on earth is 9.8ms-2. From results, it is shown that when a 0.30m and 0.60m pendulum was used, its gravitational pull is calculated to be 10.2ms-2 and 9.6ms-2. Consequently there is a percentage error of 4% and 2% respectively. Since the percentage error is less than 10%, the values are considered acceptable, however when the 0.90m pendulum was used, its gravitational pull was 9.8ms-2, which is the same value as the value of the theoretical acceleration due to gravitation on Earth.

Within the experiment, the amplitude of the displacement is kept under 7° at 5°for all pendulum measurements. Due to this the motion of the pendulum is closely related to the simple harmonic motion (Houston 2012), hence the restoring force of when the object swings back to the original position is directly proportional to the displacement of 5°. Due to this the pendulum will continue to swing back to the original launch position (Houston 2012), however factors that affects it are the length of the pendulum and the acceleration due to gravity. This controlled factor increases the reliability and accuracy of the results as if the displacement is above 7° then when the pendulum swings, there would be no restoring force, hence there would be less of a chance for the pendulum to return to the original position, and this will affect the cycle time.

Nevertheless, uncertainties were calculated for all measurements of the pendulum. For the 0.30m pendulum, it was calculated from the results that the lowest uncertainty for the acceleration due to gravitation is 8.4ms-2 and highest is 11.9ms-2. The acceleration due to gravitation from using the time from the three trials is within the range of 8.4ms-2 and 11.9ms-2. This is also the same for the 0.6m pendulum where its highest acceleration is 10.5ms-2 and lowest is 8.8ms-2, and 0.9m where its highest acceleration is 10.4ms-2 and lowest is 8.2ms-2.

Though there were some errors presented as the acceleration from the 0.30m pendulum and 0.60m pendulum did not correspond with Earth’s actual gravitational acceleration. One of the errors is believed to be parallax error, which is caused by the difficulty to determine exactly when the pendulum returned to the original launch position after a full oscillation. This error could have either increased or reduced the time recorded for the pendulum to oscillate. Thus, by increasing or decreasing the time, it affected the calculation for the acceleration due to gravity for each individual and average measurement.

To improve the experiment, a longer pendulum is to be used. This lessens the chance of parallax error; hence the oscillation time recorded and lessens the chance of random error, which also increases the precision of the data. A longer pendulum would cause the time it takes for a pendulum to cycle to be longer as time is proportional to the square root of length. A longer cycle makes it less difficult to record exactly when the pendulum return to its original launch position

Conclusion

The acceleration due to gravitation was determined to be 10.2ms-2, 9.6ms-2 and 9.8ms-2 for the pendulum measurements of 0.30m, 0.60m and 0.90m. This shows that the aim f the experiment was achieved through the conduction of the experiment. Though, the theoretical acceleration due to gravitation on Earth is determined to be 9.8ms-2, in which it was found that by using the 0.90m, the exact value could be calculated. However there were some errors involved such as the parallax error, but within all trials, the acceleration due to gravity of each individual was within the highest and lowest uncertainty range. An improvement was suggested in regards to the errors and that was to use a longer pendulum to reduce the pendulum cycle time. Overall the experiment was followed according to the method, and the result obtained had a percentage error less than 10%, hence the results are considered acceptable.

References

Ashbacher, C 2002, ‘Sir Isaac Newton: The Gravity of Genius’, Mathematics & Computer Education, vol. 36, no. 3, pp. 302-310, viewed 5 September, via Education Research Complete

Houston, K 2012, ‘The Simple Pendulum’, College Physics, vol. 1, no.1, pp.1-4, viewed 5 September, <http://cnx.org/content/m42243/latest/?collection=col11406/latest>

Appendix

Diagram 1.1

Experiment Set Up