Title

Measuring (g) using pendulums of different lengths

Abstract



Introduction

This report will attempt to find a value for the gravitational acceleration (g) in Bristol University by conducting an experiment involving the recording and examining the periods of pendulums of varying lengths. Each variant will be released with an uncontrolled amplitude. The periods of the pendulums with be measured with a human controlled stopwatch using visual observation.

With this

Theory

The period of a pendulum is dependent on length, not amplitude, so it will not matter what amplitude is used. The mass of the pendulum is also irrelevant.

Small g is not constant and is slightly different in all places around the world. This is because the Earth is not a sphere. If it was a sphere, all of the points on its surface would be equidistant from its physical centre, as wall as its gravitational centre. Earth's surface is naturally non uniform. The parts higher than average will have lower gravity than average because they are physically further away than the gravitational centre. The reverse is true for parts of the Earth's surface lower than average.

Experimental Methods

This experiment was conducted with the following equipment to the following instructions.

A length of twine that exceeds the maximum chosen pendulum length (~2.15mm diameter)
A metal ball with a hook attatched (38.51g)
A stable table that is level (~904mm tall)
A metal clamp with a loop that extends the edge of the table
A stopwatch that records to decimal places
A Lab-book and pen to record measurements
One steel meter rule

1) Tie one end of the string to the hook of the metal ball.

2) Clamp the clamp to the edge of the table, leaving room for a swinging pendulum.

3) Tie the end (that is opposite to the ball) to the loop of the clamp at any length.

4) Measure the length of the string from its fixed point on the clamp to the centre (of gravity) of the ball ten times and record it with the pen into the notebook.

5) Use the stopwatch to try to record 2 seconds thirty times and record the actual measured time into the notebook with the pen.

6) Draw a table in the lab-book in preparation to record results in.

7) Construct a pendulum (with the string, ball, clamp, table assembly) of chosen length smaller than the height of the table.

8) Release the pendulum with any amplitude, ensuring the stopwatch is started simultaneously.

9) After (m) swings, record the time elapsed into the lab-book with the pen.

10) Repeat this test a chosen number of times.

11) Repeat from step 7 for every chosen length.

12) Calculate string measuring (length) error and stopwatch (timing) error and stopwatch timing error using the recorded information from steps 4 & 5, as well as error from repeating the experiment for each length. It is crucial the calculated error from adding stopwatch error and repetition error is correctly used to calculate the error of the mean values for time^2.

13 ) Plot experimental results (length against time^2) on graph paper or digitally using error bars calculated from step 12.

14) Draw lines of maximum and minimum slope that goes through most of the points and the origin.

15) Draw a line of best fit. The slope of this line = 4pi^2/g

16) Calculate a value for g by dividing 4pi^2 by the gradient of the line of best fit.

Results

Figure (1) shows the time taken to complete one full period by pendulums of varying lengths.



Discussion
The results shown in figure (2) don't come very close to matching the projected gradient (green line) representing the standard value for (g) set out in the The General Conference on Weights and Measures (1901). With the error in these results being so low, it is unlikely this disparity was due to random error, nor is it likely that the University is under a very localised stronger gravity. It is most likely that there was a systematic error that caused these results to point to a straight line with an incorrect gradient. It is then poised that the results in this report will be of little use to future scientists performing similar experiments.

Conclusions
The value for gravitational acceleration at Bristol University was calculated from observations to be 11.21 m/s^2


References