Using a Pendulum to determine Gravity due to acceleration (G)

Aim:

To observe and calculate the rate of acceleration due to gravity by investigating the gravitational effects on the oscillatory motion of an average pendulum.

Background:

When a pendulum swings with a small angle, the mass on the end performs a good approximation of the back-&forth motion (simple harmonic motion) the period of the pendulum is the time taken to complete one single back and forth motion. This depends on just two variables length of the string and the rate of acceleration due to gravity. The mass has a very minute or no affect at all. The formula to find a period is:

Where T- is period (s) i.e. time for 1 oscillation l - is length (m) of the pendulumg - is acceleration (m/s2 )due to gravity

Apparatus:

Retort Stand Clamp Boss Head String 1- meter Mass Carrier

Mass 50g Metre Rule Stopwatch Large Protractor

Variables: Independent – the length of the string Dependent – the time it takes the pendulum to oscillate Controlled – the mass of the pendulum, angle of string from the vertical

(plumb line), amount of friction (air resistance) kept the same, retort stand kept in the same place on the bench to preserve reliability.

Safety risks: Do not use an excessive amount of weights as it could strain muscles and

cause injury to students Keep swinging pendulum clear of other people

Method:1. Set up the apparatus as shown in the diagram below.

2. Collect a mass and tie it strongly to a string.

3. Pick up the mass carries by the string and have another member of the group carefully measure 1 metre of the string starting from the base of the mass carrier.

4. Record the length of the pendulum. Attach the string to the clamp really tightly and ensure that you still have 1 meter from the top to the bottom of the string.

5. Ensure that the vicinity is free of any obstructions to the swinging pendulum.

6. If necessary place a g-clamp or excess weight on the retort stand to stop the retort stand from absorbing the motion energy of the pendulum by vibrating.

7. Reset all stop watches and gently move the pendulum from equilibrium to a measured distance of 10º or less using a large protractor. Ensure the angle of deviation from the vertical is measured properly and is kept the same throughout all trials.

8. Carefully release the mass from the deviated angle and allow it to swing for 2-3 swings and lose some of the vibrations that may have been transferred.

9. Activating the stopwatch as the string oscillation commences a new period.

10. Continue timing the pendulum until it has moved through 10 complete oscillations (periods) and record the times.

11. Repeat steps 3 through 9 a total of 4 more times. However before each new set shorten the string by 20 cm of its length. Ensure that the deviation angle is controlled for constancy through all trials

String

Counter Weight

Clamp

Retort Stand

Mass

Note: Ө <10º to avoid angular displacement.

Ө

Horizontal support

Length

Mass

Results:

Results attained form different lengths and 4 trials

Length: 1m

Trial Time for 10 oscillations(s)

Period (s)

AveragePeriod2 (S2)

1 19.94 1.994 3.9362 19.84 1.9843 19.75 1.975

Analysis:

Length: 0.6m

Trial Time for 10 oscillations (s)

Period (s)

AveragePeriod2 (S2)

1 17.81 1.781 3.1542 17.75 1.7753 17.72 1.772

Length: 0.8m

Trial Time for 10 oscillations (s)

Period (s)

AveragePeriod2 (S2)

1 15.50 1.550 2.3622 15.38 1.5383 15.22 1.522

Length: 0.4m

Trial Time for 10 oscillations (s)

Period (s)

AveragePeriod2 (S2)

1 12.56 1.256 1.5552 12.38 1.2383 12.47 1.247

To Find g:

T = 2π√(l/g) à T/2π = √(l/g)

T2/4π2 = l/g

gT2/4π2 = l

g = 4π2l/T2

Determining Gg = 4π2l/T2

g = 4π2(1/3.936)

g = 10.03008577

T2= 3.936L = 1.0 meter

Determining Gg = 4π2l/T2

g = 4π2(0.8/3.154)

g = 10.01354917

T2= 3.154L = 0.8 meter

Determining Gg = 4π2l/T2

g = 4π2(0.6/2.362)

g = 10.0283872

T2=2.362L = 0.6 meter

Determining Gg = 4π2l/T2

T2= 1.555L = 0.4 meter

g = 4π2(0.4/1.555)

g = 10.15521996

Final Average of G:

g1 + g2+ g3 + g4 = …….

10.03008577 +10.01354917 +10.0283872 +10.15521996

= 10.05681053

= 10.06 m/s2 (2 dec.pl.)

Finding g using the graph

Sub m into eqn: g = 4π2 x 1/m

Run

Run

Rise

Length (m)

T2 (s2)

m = rise/run

Gravity as determined form the graph

m = (3.963 – 1.555) / (1 - 0.4)

m = 3.968333…

g = 4π2 x 1/m

g = 4π2 x 1/3.968333

g = 9.94836227

g = 9.95 m/s2 (2 dec.pl.)

Gravitational acceleration was found to be approximately 10.06 m/s2 from the result calculations and 9.95 m/s2 from graphical solution. These values were approximately 1.5% and 2.6% respectively off the accepted value of 9.8m/s2.

Through observation of the graphical results and the line of best fit, the results were found to be very accurate and precise. The line of best fit passes through each point and is only just off the origin. These are the desired results for such an experiment.

Conclusion: By observing and calculating the rate of acceleration due to gravity by investigation the effects of gravity on an average pendulum, it was found that the calculated gravitational acceleration was about 10.005 m/s2. This value was not far off of the accepted value of 9.8 m/s2.

Discussion:The independent variable in this investigation was the length of the string and, therefore, the length of the pendulum this is only if the dimensions of the mass are kept constant which in this case were. The reason for starting the experiment from 1 m with 0.2 m in between was to increase the accuracy of measurements and in turn minimise error. Using shorter lengths was not a good idea because shorter pendulums have shorter periods. Since measurements of period were taken with a stopwatch by a timekeeper, the shorter the periods would have been more difficult for the timekeeper to make accurate judgments o when to start and stop. Using the longest strings is very practicable and means that this source of error was reduced in this investigation.

The second dependent variable in this investigation was period of oscillation. For a pendulum in simple harmonic motion with a small deviation angle, period of oscillation depends only upon the pendulum length and the acceleration due to gravity. The reason for timing 10 oscillations, rather than just one, was to eliminate the errors in judgment associated with panic and mad scrambles during short time frames. Prolonging the oscillations meant that the timekeeper was able to better anticipate the point of closure and, hence, take a more accurate reading of time. A possible source of error in this procedure, however, lies in the division of each recorded time by 10. This was done on the assumption that period of oscillation remains constant for 10 full oscillations, when, in reality, it would decrease over time (since the pendulum would lose momentum through interactions with forces retarding its motion, including air resistance).

The value for gravitational acceleration calculated in this experiment differed slightly from the theoretical value of 9.80ms-2 published in each of the below texts. One possible reason for this deviation lies in the levels of accuracy of the measuring instruments used. The limits of reading of the instruments, and of the rule and stopwatch, in particular, were a limitation in this investigation, and a barrier to achieving results of utmost exactness and, hence, a conclusion of utmost reliability. Substitution of measuring apparatus of higher levels of accuracy would have improved the validity of the conclusion through minimising absolute errors in both collected and calculated data.

Gravitational acceleration was both a calculated, and a controlled variable in this investigation. The formula above works on the assumption that acceleration due to

gravity is a constant. However, it is known that gravitation acceleration changes with such factors as altitude, crustal density and position on the Earth’s surface. Although this was known, when the length of the string was measure, the distance between the mass and the ground changed. To improve results, no change in string length should have been made without adjusting the boss head and clamp so as to keep the distance between the mass and the ground constant for all trials.

Another reason for the discrepancy between the true and experimental values for gravitational acceleration could have been the failure of the investigation to adequately account for the error ranges of measuring instruments in both calculations and the graphical representation. To eliminate this error source, these ranges could have been factored into calculations involving T, T2 and l, giving more exact values of g and bringing greater validity to drawn conclusions. Also, instead of simply taking the average of the 4 values of g as the definitive value, an allowance for error could have been made by determining the greatest residual from the arithmetic mean and expressing the final value as a range, rather than a definite figure. This would have had the added advantage of showing clearly the level of accuracy of the investigation and, hence, giving a truer indication of the reliability of the conclusion.

A possible source of error, and a possible cause for the difference between the value of g calculated in this experiment and the theoretical value, lies in the variations in gravitational acceleration that relate to geographical position. Depending on the thickness and density of the Earth’s crust, proximity to the Earth’s poles and the magnitude of centrifuge forces at any one point on the Earth’s surface, the value for g calculated in this experiment could have deviated by as much as 0.032ms-2 due to factors beyond direct control.

Also contributing to the stated discrepancy could have been inherent faults in the apparatus used, including weak and/or worn components of the boss head, clamp, mass and/or retort stand, as well as frailty of the string, or even a weakening of an otherwise strong string through repeated use. Solutions to this source of error include replacing the string with a fresh length before each new trial and carefully examining and replacing other apparatus where, and when, necessary.

Another reason the validity of conclusions may have suffered could have been the intervention of humans in both the data collection, and the data analysis process. Both systematic and accidental errors, including those related to parallax, arising from human involvement would have had a negative impact on the reliability of gathered data, the accurate analysis of that data, and the validity of the drawn conclusion. Replacing humans with artificial intelligence in the form of robots and/or computers in the areas of data collection and analysis would have rectified this error source and improved the reliability of the investigation as a whole.

Each time the pendulum is brought from equilibrium back to its extreme of motion before release, it is critical that no, or, at the very least, little tension is lost from the string. By supplying flexion to the string, the mass carrier is given additional potential energy on top of the weight force already being exerted. This means that, on release, the pendulum will have additional and unwanted forces acting on it, resulting in further reaction forces, impulses through the string and the disturbance of harmony motion. This could lead to inaccurate results and an unreliable conclusion.

The experiment was reasonably reliable as tests for each length were conducted multiple times. To increase this reliability, many groups executed the same investigation, all retrieving similar results. The experiment was also very valid as there were many variables being controlled, but could have been more valid by improving such variables including the distance between the ground and the string.

While conducting the experiment, it was observed that as the string completed the first few oscillations, it began to spin in an anti-clockwise circular motion. It performs these motions due to the spinning of the earth. The pendulum follows the path of the Earth and its gravitational force.