Calculating Free Fall Acceleration
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Published: Tue, 09 Jan 2018
Introduction
A research by Heckert (2010) shows in 1600s, the famous physicist Galileo . Galilei found the swinging motion of a large chandelier in the Pisa cathedral. He began to seriously analyse the chandelier, and recorded the time the light took to swing. In the 16th century, there was no stopwatch so that Galileo timed the swing by pulse. In addition, he was the first European to really study this phenomenon and he discovered that their regularity could be used for calculate the local gravity.
For Galileo his pendulum was the light but generally speaking a pendulum can be defined as a body suspended from a fixed point which swing freely by the motion of gravity and momentum. It is used to regulate the movements of clockwork and other machinery.
In its simplest form and avoiding the math there are three parts to the basic laws of a pendulum. First the time for each oscillation is depending on the length of the strings. In addition, mass of the bob does not affect the motion at all. Second, a pendulum’s horizontal speed is the same as the vertical speed would be, if the bob had fallen from its highest point. Thirdly, the square of period of the bob is inversely proportional to free fall acceleration and the square of period of the body is proportional to length of the pendulum
The background definition and the laws of a pendulum can be used to calculate the free fall acceleration. Using a simple gravity pendulum like Galileo’s Pendulum System, I would like to show how to find the best ways in order to test free fall acceleration.
Methods
1. Experiment equipment:
 Protractor
 Steel Bob
 Stopwatch
 Vernier Caliper
 Iron Support Stand
 Meter Ruler
 Inelastic String
2. Apparatus setup Figure11
Figure11 shows that iron support stand was put beside edge of test desk in case the height of stand was shorter than the length of test string. Next, the steel ball was hung by an inelastic string and the iron support stand was used to support the weight of steel ball. Last, the clip was clamped to the string in order to keep a constant length. At the same time, the bob swing in a vertical surface which parallels the stand.
3. Procedures
First of all, the simple pendulum was made up by hanging a bob from the top of stand and the bob was released in a constant height, then protractor was used to control the degree between 5○ and 15○ to normal line. Secondly, pendulum would begin to oscillate in vertical surface in a regular action, and then the stop watch was used to record the time of each swing. Finally the most important data which describes this oscillation is period and we did different types of test by different length of string, like 30cm, 45 cm, 60 cm, 75 cm, 90cm, 105 cm, and 120 cm.
Results
Table of result
Experiment times 
Length of string(cm) 
Trials: 1 
Trials: 2 
Total Average period 
Oscillationtimes 
Average period of each swing 
T2(second square) 
Time taken for one complete Oscillation(seconds) 

1 
30cm 
56.60s 
56.50s 
56.55s 
50 times 
1.13s 
1.28s2 
2 
45 cm 
68.60s 
68.50s 
68.55s 
50 times 
1.37s 
1.88 s2 
3 
60 cm 
79.00s 
78.90s 
79.00s 
50 times 
1.58s 
2.50 s2 
4 
75 cm 
87.60s 
87.90s 
87.75s 
50 times 
1.76s 
3.08 s2 
5 
90 cm 
96.05s 
96.00s 
96.05s 
50 times 
1.92s 
3.69 s2 
6 
105 cm 
104.00s 
104.00s 
104.00s 
50 times 
2.08s 
4.33 s2 
7 
120 cm 
110.50s 
111.00s 
110.75s 
50 times 
2.22s 
4.91 s2 
Table1.1
Table1.1 shows the data of 7 experiments using different length of string and how the data changed, as the length of string was increased; the period of each oscillation was increase as well.
L is the distance from the frame of the stand to the center of the mass; the length includes the radius of ball. The period of oscillation is the time required for the pendulum to complete one swing. For one complete swing, the steel ball must move from the left to the right and back to the left. T2 can be understood as the square of the period of oscillation, the equation below shows how T2 was calculated.
Square both sides:
T2= 4 × π2 × (L/g) T2 = L × (4 × π2 ÷ g)
Multiply both sides by g
g × T2 = 4 × π2 × L
Divide both sides by T2
Discussion and Analysis
The results of experiment show the relation between T2 and length of string. To turn to discuss the results it is important to understand some key ideas, there are controlled variable, experimental variable, error and uncertainty.
Firstly, according to Science Buddies(2009) said that a controlled variable can be defined as the factor which is unchanged or kept constant to prevent its effects or error on the outcome. It was verified the behavior of the relationship between independent and dependent variables. The factors which can be regarded as controlled variable were steel ball, oscillation times; the angle of each swing and the height when the steel ball was released. An answer from wiki (2009) the definition of experimental variables is the variable whose values are independent of changes in the values of other variables. Experimental variable in this experiment is the length of string.
According to dictionary the error can be defined as a deviation from accuracy or correctness. And the uncertainty means that the lack of certainty, a state of having limited knowledge so that it is impossible to exactly describe existing phenomenon or future outcome confidently.Errors were caused by any individual who could be affected by many factors. Such as before we measure the length of string, we need to measure the radius of ball by vernier caliper in case the string is shorter than actual length. Secondly, we need to take care of how much oscillation times we did. Thirdly, we need to keep the pendulum swing in a same surface in case the extra energy was wasted. At last, taking more time measurements of experimental variable which is length of string may be more accurate average for each trial.
Find two point from the graph A(x1, y1) B(x2, y2), use the formula：
(y2y1)/(x2x1) the result of gradient is 4.03.
The table shows the results of free fall acceleration
Gradient(T2/L) 
4.03 

Calculate data in using formula 

△G 
9.79ms2 

Confines of Error 
0.22% 
Table21
To summarize the weakness that is error and uncertainty and calculating the acceleration of gravity to within 5%, and table 21 shows that the experiment obeys the allowable confines. Confines of Error were calculated by the difference between actual gravity and what I got, and the values were divided by the actual values.
Conclusion
To sum up, the calculation of Galileo that free fall acceleration from the formula, this can infer the result of free fall acceleration. I need to compare the calculation of Galileo which free fall acceleration should be 9.81ms2. In fact, a gravity pendulum is a complex machine, depending on a number of variables for which we are ready to adjust.
In addition, firstly we try to understand the method that Galileo did in 1600s, and making a plan to have a complete the system. Then form the data I found some different values about gravity, and the factor to influence the values. The main factor is that the different length of string influence the period instead free fall acceleration, the period square and length have a constant ratio to calculated the acceleration.
Turning to Dohrman, P (2009) it can be argued that the factors which influence the fact are length of the string, period of each cycle by using those two factors we can get the local gravity. All above those factors can influence the values of free fall acceleration, and we got the less number than actual values. I need to take care of them and have an improvement. For instance, first difficulty is that measuring the length is deciding where the centre of the bob is. The uncertainty in determining this measurement is probably about 1 mm. Secondly, the stopwatch measures to 50 of oscillation although the overall accuracy of the time measurements may be not certain. According toDohrman (2009) the human reaction time to start and stop the watch has a maximum range of 0.13 seconds and the average is0.7. Finally, 9.79ms2 was calculated by the gradient and the formula in part of result.
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