# Measuring Acceleration Due to Gravity Using a Simple Pendulum

### A-Introduction:

1. Objective: The main purpose of the experiment is to measure the acceleration of gravity as accurately as possible, then, gain more knowledge on acceleration and pendulum. The most important aim is that improve the ability of study and experiment.

2. Background: A pendulum is that a mass suspended from a fixed point so that it can swing in an arc under the influence of gravity. The time period of a pendulum is constant. The changing of the mass of a weight can't affect the period. But it can be changed by decreasing or increasing the length of pendulum. Because of their constancy, some special pendulums, such as Foucault Pendulum are sued to measure the value of the acceleration of gravity. The earliest uses of a pendulum was seismometer device in the first century. Its author is Han Dynasty Chinese scientist Zhang Heng. Its function was to sway and activate one of a series of levers after being disturbed by the tremor of an earthquake far away. But the first to study the properties of pendulums was Italian scientist Galileo Galilei , beginning around 1602.

### B-Method:

1.The apparatus of the experiment: stand, bob, thread, split cork, rule, stop watch.

(Picture 1)

### 2. Procedure:

Set up the pendulum as shown as picture 1 and picture 2. We known that the equation of time period of pendulum is T=2π(l/g)1/2. We squaring the both side of the equation and then re-arranging the equation gives g=T2/4π2l. Hence, we just need measure the length and time period of the pendulum.

Firstly, using the ruler measure the length of the pendulum. Secondly, allow the pendulum to swing through a small angle and measure the time for 10 complete oscillations then find the time period T by dividing the time measures by 10. Thirdly, repeat for 5 more different lengths of the pendulum between 0.2m and 1.2m. Lastly, analyse the data of you measured and find the value of g by using the equation g=T2/4π2l.

(Picture 2)

### 3.Precautions for accuracy:

(a)Select a small, light and not easy to stretch line. Select a high density and small diameter metal bob, the diameter not more than 2cm preferably.

(b)The top line of pendulum can not be arbitrarily suspended in the metal trap, should be clamped in a split cork in order to avoid occur the phenomenon of the length of pendulum changing and cycloid downing.

(c) Note that the angle between cycloid and vertical not more than 10 °.

(d) When the pendulum swings, make sure that the track of the bob moving at the same vertical plane, do not form a conical pendulum.

C-Data gathered:

The following tables show that we measured data the length of pendulum and time for 10 oscillations.

No. of trails |
Length l (m) |
Time for 10 oscillations (s) |
Time Period T (s) | |

Measured |
Used | |||

1 |
0.20 |
8.8 |
8.8 |
0.88 |

8.8 | ||||

2 |
0.40 |
12.1 |
12.2 |
1.22 |

12.4 | ||||

3 |
0.60 |
15.3 |
15.2 |
1.52 |

15.1 | ||||

4 |
0.80 |
17.6 |
17.6 |
1.76 |

17.6 | ||||

5 |
1.00 |
19.7 |
19.6 |
1.96 |

19.5 | ||||

6 |
1.20 |
21.6 |
21.6 |
2.16 |

21.6 |

(Table 1)

### D- Processing:

We known that the equation of time period of pendulum is T=2π(l/g)1/2. We squaring and then re-arranging the equation gives . Suppose X=T2 and Y=4π2l , the equation 4π2l=gT2 can be written to Y=gX. Hence, the graph of the equation is a straight line and the gradient of this line is g.

So we can obtain the following table and graph:

No. of trails |
Length l (m) |
Time Period T (s) |
X=T2 (s2) |
Y=4π2l (m) |

1 |
0.20 |
0.88 |
0.774 |
7.90 |

2 |
0.40 |
1.22 |
1.49 |
15.8 |

3 |
0.60 |
1.52 |
2.31 |
23.7 |

4 |
0.80 |
1.76 |
3.10 |
31.6 |

5 |
1.00 |
1.96 |
3.84 |
39.5 |

6 |
1.20 |
2.16 |
4.67 |
47.4 |

(Table 2)

### E-Discussion:

From the date we can see that the experiment value of g is 10.25m/s2 which is beyond the theoretical value of g which is 9.81m/s2. The occur of this problem because of the following reasons. Firstly, the measuring value of length of pendulum is not accurate. the real length of pendulum is the length of thread from the split cork to bob adding the length from the top of the bob to its center of gravity. But I miss the length from the top of the bob to its center of gravity. Secondly, the reaction of recorders who record the time when they saw the bob moving probable delay. In the equation of 4π2l=gT2 , if Y(4π2l) is constant, the relationship of X(T2) and g is negative correlation. Due to the delay of recorder's reaction, the measured value of T will less than the real time period. Also the value of X we obtain less than its real value. Hence, the value of g we gain will more than 9.81m/s2. Lastly, the error which come from the process of map-making.

### F-Conclusion:

The experiment tells us how to find the value of acceleration of gravity. If we have more perfect experiments and more time, the result will be more exact. Through the experiment I know more about the subject. I also learned the method of using experiments to prove the theory. I learnt if want to get exact results you must becareful and patient. I have some suggestion to this experiment. Choose a metal ball which has a hole at the top and measure the radius of ball by using vernier caliper. So, the pendulum length will more exact. measure the time for 20 complete oscillations and choose more advanced calculagraph, we will improve the accuracy of values of time.

Above all, I think that this experiment is successful though we obtained the experiment value of g is beyond the theoretical value of g.

### G-References:

Admin (2010) 单摆侧当地的加速度, Retrieved April 5, 2010, from http://www.8bu.net/gaoerwuli/15712.html

Simple Pendulum, Retrieved April 5, 2010, from http://www.tutorvista.com/content/physics/physics-i/measurement-and-experimentation/simple-pendulum.php ( Picture 1 and Picture 2)

Simple gravity pendulum, Retrieved April 5, 2010, from

http://encyclopedia2.thefreedictionary.com/Simple+gravity+pendulum

Pendulum, Retrieved April 5, 2010, from http://en.wikipedia.org/wiki/Pendulum

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