Aim:

To investigate the relationship between the length of pendulum and the time it takes.

Theory:

Hence the gravitational acceleration “g” can be calculated by this equation: T=2πLg

T=2πLg

(T the period in seconds, l the length in meter and g the gravitational acceleration in m/s2)

Method:

As stated earlier in the introduction, it was decided to measure the time for 10 complete swings, in order to reduce the random errors. These measurements would be repeated two more times, and in total five successive lengths were used. We start from one metre, and decreasing by 20 cm for each following measurement, and the mass of metal bob doesn’t change all the time. According to pendulum is periodic. We will push it and record the period time of swinging (T) when it comes back. Then we can draw a diagram with data by Excel. The experimental set-up was equal to the diagram, shown in figure 1

Figure 1: Diagram of the set-up for this experiment

Results:

Choosing the data randomly then forming a diagram and a graph:

Length (m)| Time| Period (s)| Period2 (s 2)|

0| 0| 0| 0|

0.2±0.001| 9.12| 0.91±0.01| 0.83±0.02|

0.4±0.001| 12.71| 1.27±0.01| 1.62±0.02|

0.6±0.001| 15.62| 1.56±0.01| 2.44±0.03|

0.8±0.001| 17.9| 1.79±0.01| 3.20±0.03|

1±0.001| 19.81| 1.98±0.01| 3.92±0.04|

1.2±0.001| 21.72| 2.17±0.01| 4.72±0.04|

Sources of errors: The error in time was 0.01s because the watch is digital and measured to 0.01s. The error in reading ruler is 0.5mm, as the ruler is analogue and its smallest scale is 1mm. But the position of the mass was not clear so the error was made 1mm in length.

The graph of period versus length is as below:

We can see that the trend line of period against length is not a straight line.

The graph of period2 versus length is as below:

From the graph of Period2 versus length as above, we can see the gradient of period2 against length is 3.94s2/m, so the best-fit value of g is 9.92s2/m.

Discussion:

From the table and the equation, we can calculate the value of gravitational acceleration “g”: When L=0.2m, T=0.91s. Hence g= 9.53m/s2

When L=0.4m, T=1.28s. Hence g=9.63 m/s2

When L=0.6m, T=1.56s. Hence g=9.72 m/s2

When L=0.8m, T=1.79s. Hence g=9.84 m/s2

When L=1.0m, T=1.98s. Hence g=10.05 m/s2

When L=1.2m, T=2.18s. Hence g=9.95 m/s2

Then we can see

The maximum value of g is 10.05 m/s2

The minimum value of g is 9.53m/s2.

The best-fit value of g is 9.92s2/m, considering the maximum and the minimum value, we get the experimental value of g: g=9.92±0.4s2/m. It is similar to the known value 9.8s2/m. Comparing our calculated value for the gravitational acceleration ‘g’ with the known value, we can found some different. The systematic error and human error couldn’t be avoided in our experiment, but we can reduce these errors as many as we can. Maybe we should have measured the time for 20 swings, instead of 10 swings, which would also reduce our uncertainty in time. Also we need control the metal bob came from the same position or we shuold do this experiment under the vacuum condition, to reduce the effect of air resistance on this experiment.

Conclusion:

1. To a pendulum, the square of its period is proportion to its length. 2. The experimental value of g is 9.92±0.4s2/m. It is similar to the known value 9.8 s2/m. Allowing for experimental errors, the result was expected.

REFERENCES

The NIST Reference on Constants, Units and Uncertainty [Online] Available at: http://physics.nist.gov/cgi-bin/cuu/Value?gn

[Accessed 23 March 2013]

Practical Physics. 2009. The Swinging Pendulum [Online]

Available at http://www.practicalphysics.org/go/Experiment_480.html [Accessed 23 March 2013]