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Experiment to determine gravity from a spring using analogue techniques

Extracts from this document...

Introduction

Experiment to determine gravity from a spring using analogue techniques

The aim of this first experiment is to examine simple harmonic motion exhibited a mass on a spring. Using data recorded in doing this, the spring constant for each spring can be calculated along with a value for gravity.

In the first part of this experiment, the relationship between the period of the oscillations of the spring and the mass of the spring is observed. The period of oscillation of mass on a spiral spring depends on the mass on the spring and the spring constant of the spring. This is given by:

image05.png

Where m is the mass on the spring and k is the spring constant of the spring. Since the period can be observed, and the mass on the spring is known, this part of the experiment is concerned with calculating k, the spring constant for each of the springs used.

The spring constant is different for every spring, and is defined as the mass needed to produce a unit extension of the spring (ref. 6). This is calculated by placing differing masses on the spring, extending the spring a certain distance from its equilibrium position each time and timing the time for 10 oscillations of the spring to occur. This is done by using an analogue stopwatch and a ruler to ensure that the distance extended from the equilibrium position was the same each time. The graph of period squared against mass can then be plotted.

...read more.

Middle

3

6.8

4

6.7

Average time for 10 oscillations(s)

Random uncertainty (s)

2.90

0.05

3.58

0.025

4.20

0.05

4.70

0.05

5.05

0.05

5.45

0.025

5.80

0.05

6.05

0.025

6.43

0.025

6.73

0.05

Taking the average time for 10 oscillations, calculating the period and the period squared:

Mass (kg)

Time for 10 oscillations (s)

Period (s)

Period squared (s2 )

0.01

2.900

0.290

0.084

0.02

3.580

0.358

0.128

0.03

4.200

0.420

0.176

0.04

4.700

0.470

0.221

0.05

5.050

0.505

0.255

0.06

5.450

0.545

0.297

0.07

5.800

0.580

0.336

0.08

6.050

0.605

0.366

0.09

6.430

0.643

0.413

0.10

6.730

0.673

0.453

So the graph of period squared against mass is:

image07.png

Since this graph is a straight line that should go through the origin, it can be seen that the period of the spring squared is directly proportional to the mass placed upon it.

image08.png

The equation for this graph can now be compared to image09.png to gain a value for the spring constant of this spring:

image09.png

image06.png

Since image10.pngis the y value of the graph, and m is the x value of the graph, it follows that:

image11.png

The spring constant, k, can be found by substituting the value of the gradient of the graph into this equation:

image12.png

This is the value of the spring constant of the spring.

Uncertainties

To Calculate the Uncertainty in k

In order to calculate the uncertainty in k, a parallelogram is drawn around the extreme upper and lower points of the trendline, and the corner points of the parallelogram are recorded. The uncertainty in the gradient, and thus k, can then be calculated according to the equation:

image13.png

Where m(AC) and m(BD) are the gradients of the diagonals of the parallelogram and n is the number of points in the graph.

A(0.01, 0.10)  B(0.10, 0.46)  C(0.10, 0.445)  D(0.01, 0.08)

image14.png

image15.png

image16.png

This uncertainty will be the same for k, because the other component’s involved it it’s calculation are all constants.

Therefore:

image17.png

To Calculate Uncertainty in Each Point

The main uncertainties are the random uncertainty, reading uncertainty and the calibration uncertainty. These can be combined using the equationimage18.png, to give a total uncertainty for the period of the spring. This is then calculated as a percentage, and doubled, since the graph uses the period squared, and the total uncertainty for each period squared value can be calculated.

A sample uncertainty calculation is shown below.

Mass = 0.01kg

Random = ±0.05s

Reading = ±0.05s

Calibration = 2% x 2.9 = ±0.058

image19.png

As a percentage of 2.9, this is 1.1%.

This is doubled, to 2.2% uncertainty in period squared, so:

image20.png

All uncertainty calculations were done as shown above, and the results were:

Mass = 0.02kg

image21.png

Mass = 0.03kg

image22.png

Mass = 0.04kg

image23.png

Mass = 0.05kg

image24.png

Mass = 0.06kg

image26.png

Mass = 0.07kg

image27.png

Mass = 0.08kg

image28.png

Mass = 0.09kg

image29.png

Mass = 0.10kg

image30.png

Because all these uncertainties are very small, the error bars on the graph are too small to see.

Spring 1 – Part 2

Mass (kg)

Extension (m)

0.005

0.002

0.010

0.006

0.015

0.011

0.020

0.015

0.025

0.020

0.030

0.024

0.035

0.029

0.040

0.032

0.045

0.037

0.050

0.042

0.055

0.046

0.060

0.051

0.065

0.055

0.070

0.060

0.075

0.064

0.080

0.068

0.085

0.071

0.090

0.076

0.095

0.081

0.100

0.086

...read more.

Conclusion

Using LINEST, the value for the gradient is: image51.png

This is combined for the uncertainty in g.

image52.png

So the value for g is:

image53.png




Discussion

Conclusion

This experiment has shown the relationship image54.png for a mass m on a spiral spring with a period of oscillation T. Values for gravitational field strength were also calculated at image39.png and image53.png.

Evaluation

This experiment can be seen as somewhat successful, with the values for gravitational field strength being close to the generally accepted value of 9.81Nkg-1.

The reading disk supplied was an accurate way of determining the distance extended, but the disk often slanted, which could affect measurements. In order to counter this, the distance was read at the same point on the disk each time, so this effect could be negated.

The stopwatch supplied was an analogue one, which only measured in increments of 0.1s. In future, a digital stopwatch would be used, which could measure in increments of 0.001s, in order to increase the accuracy of the time and reduce the uncertainty.

Due to the analogue stopwatch being used, the reaction time when pressing the stopwatch could interfere with the actual time, thus interfering with the results. There is no way this could be improved, as reaction time would vary each time, and this could be why the values obtained for gravitational field strength were further out than expected. In order to combat this, a digital way of determining the time could be used, and this is done in experiment 2.

Overall, this experiment can be seen as successful with problems overcome. Because the values for g were as close as could be expected due to the uncertainty in the reaction time, this experiment can be considered a success.

...read more.

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