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Calculating the value of "g" (Gravitational field strength) using a mass on a spring

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Introduction

Calculating the value of “g” (Gravitational field strength) using a mass on a spring

Gravity affects all things that have mass and therefore must affect how much a mass placed on a spring will extend.  Measuring the time period and extension of a mass on a spring for vibrations should enable us to calculate a value for g.  Using the following formula will help us to do this:

Formula 1

T=2π√m/k

g (gravitational field strength) affects the spring constant – k in the formula F=ke and because F = weight = mg.  Therefore mg = ke and m/k = e/g.

We can now change formula 1 to the following:

T=2π√e/g

If we rearrange the above formula so that the subject is T2 we should get the formula below:

T2 = 4π2 e

         g

Measuring T would allow us to calculate T2 (The time period – to calculate measure the time it takes for a certain number of oscillations and then divide it by the number of oscillations) and e would allow us to plot a graph and, according to the formula if we take the gradient of the line of best fit it will be equal to:

 4π2

  g

We can then work out g, the gravitational field strength.

g=      4π2 _

     Gradient

...read more.

Middle

450 g

Graph information

T2 (seconds)

Extension (mm)

Reasons for choice of design using preliminary work

I completed some preliminary work before doing my actual experiment.  The preliminary work allowed me to choose the type of spring, the type of mass and the range of masses and amplitude and number of oscillations that my experiment should have.

Spring

Mass

Amplitude

Result

Short

100g

50mm

Works

Short

700g

50mm

Spring reached elastic limit

Long

100g

50mm

Mass left the spring

Long

700g

50mm

Works

Short

50g

60mm

Mass left the spring

Short

50g

50mm

Works fine

Short

50g

40mm

Works, but not enough countable oscillations produced

Short

450g

50mm

Works and enough (more than 10) oscillations produced

Spring – There were two types of spring that I could choose from – a long thin spring (average 5mm diameter, 100mm length) or a short, wide spring (average 11mm diameter, 21mm length).  I chose the short, wide spring because when the masses were placed on the longer spring they were more prone to falling off and this may have been a safety hazard.

Mass – There were two types of mass that I could have used – 100g masses and 50g masses.  I decided to use the 50g masses because after placing seven 100g masses on the short spring it stretched so much that when I allowed it to oscillate it reached its elastic limit and failed to oscillate with simple harmonic motion.  I could place nine 50g masses on the spring (450g) and the spring still oscillated with SHM and did not reach its elastic limit.

Range of masses – I tested the highest and lowest masses to reach a conclusion about the range of masses that I was going to use.  I tested to see whether the masses made the spring reach its elastic limit at the highest mass and whether the mass fell off the hook at the lowest mass.  I decided that the range of masses from 50g to 450g would give me enough results to plot an accurate graph with and 450g did not cause the spring to reach its elastic limit.

Number of oscillations – I chose the number of oscillations to be 10 because the more oscillations, the more accurate the result will be.  10 Oscillations were not too much because I originally tried 15 oscillations.  15 oscillations with 50g mass did not work because the oscillations stopped before 15 oscillations had passed, I tried 10 oscillations and this seemed to work.

Amplitude of oscillations – The amplitude of oscillations must be large enough to keep the SHM going for at least 10 oscillations, I tried quite a range of oscillations but decided on 50mm because above that was too much and the mass sometimes left the spring, causing a safety hazard and below that made the oscillations too hard to count, as they went too fast and this would have made me lose accuracy in my results.

Safety glasses must be worn as a precaution, in case the weight hook, along with the weights falls off in the middle of a cycle.

Results

Original length of spring (mm)

Original length of spring repeat (mm)

Original length of spring repeat (mm)

Average original length of spring

(mm)

20

21

21

21

Mass (g)

Loaded length of spring

(mm)

Loaded length of spring repeat 1

(mm)

Loaded length of spring repeat 2

(mm)

Average loaded length of spring

(mm)

50

38

36

37

37

100

58

59

58

58

150

81

80

80

80

200

103

104

103

103

250

124

124

125

124

300

144

149

148

147

350

165

167

170

167

400

193

189

188

190

450

202

201

201

201

...read more.

Conclusion

There may have been experimental errors in the measurement of the length of the extension:

1.0

103

=0.97% error

The total error for my experiment is 3.11 + 0.97 = 4.08%

From my experiment I calculated that g was 9.4N/Kg.  4.08% of 9.4 = 0.4

Taking this into account the actual value for g could be         9.4 ± 4%

                                                               =         9.4 ± 0.4

This gives a range from 9.0N/Kg to 9.8N/Kg  

The real value of g is 9.8N/Kg

This value is some way off my original, predicted value but this may have been because I didn’t think that my sources of error would be so significant.  The total error calculated above may not be the real total error, as I haven’t taken into account reaction times or other systematic errors.

The most significant measurement was the time period as this had to be squared, therefore the most significant error was the time period because squaring an error doubles it.

The value that I got for g may not have been accurate because of the % errors and reaction times and so may not be a very reliable result.

To help minimise errors, an electronic recording method to record the results would have increased the accuracy, because of the lack of reaction time.  A stopwatch with increased accuracy may have also helped.

...read more.

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