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# Experiment to determine gravity from a spring using digital techniques

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Introduction

## Experiment to determine gravity from a spring using digital techniques

The aim of this experiment is to look at the relationship between the mass of a mass on a spring and its simple harmonic period when it is extended then released. This should theoretically follow the relationship:

Which is in the form y=mx. This experiment will examine the straight line proportionality between the period squared of the SHM and the mass on the spring.

This will be done by varying the mass on the spring, extending the spring a certain distance, and releasing the spring. The period of this oscillation is determined and this is repeated for different masses. From this, a graph of period squared against mass can be plotted, which should exhibit the straight line proportionality as shown above.

This experiment will then use a simple rearrangement of Hooke’s law, to  to determine a value for gravitational field strength, which will then be compared to the accepted value of 9.81Nkg-1. To do this, the spring will be loaded with different masses, and the extension of the spring noted. A graph of mass against extension is then plotted, and from this a value for gravitational field strength can be calculated.

## Procedure

Apparatus

• Stand
• Motion sensor
• Computer with datastudio installed
• Slotted masses and mass holder
• CD
• Pointer
• Half metre stick
• Balance accurate to 1g
• Lab jack
• Spring

Part One

• The experiment is set up as shown.
• Datastudio is opened on the computer, and “create experiment” is clicked.
• In the “create experiment” window, the motion sensor is selected.
• In the measurement window, “Position, Ch2 & 2 (m)” is selected.
• In the “display” window, the “graph” selection was double clicked to set up a graph of the results.
• The motion sensor is placed on a lab jack on top of a stool pointing upwards towards the mass on the spring. Instead of relying on the motion sensor sensing the distance from the bottom of the mass holder, a CD is attached to the mass holder, in order to increase the area that the motion sensor can sense.
• The mass is displaced 5cm and allowed to oscillate, while the the “start” button is pressed. After at least 11 oscillations, the “stop” button is pressed to end recording.
• The “smart tool” icon was selected, and “smart tool” was used to find the times for the 1st and 11th peaks. In order to calculate the period of the motion, the time for the 1st peak was taken from the time for the 11th peak and this was divided by 10.
• The mass on the spring was weighed, including the slotted masses, the mass holder and the CD.
• This was repeated for different masses, each time weighing the masses before carrying out the experiment, in order to ensure that the mass values are accurate.
• A graph of period squared against mass was plotted and from this the spring constant of the spring could be calculated.

Middle

Part 2

• The equilibrium position of the spring is recorded when there was no mass on it, and this was taken as the 0 position.
• 50g of mass is put on the spring and the extension of the spring is recorded. The mass holder and pointer do not need to be weighed, because they were already on the spring when the equilibrium position was recorded. The mass of the mass was also weighed, to keep the results as accurate as possible.
• This process was repeated with other masses, in 50g increments, with the total mass on the spring being weighed each time.
• From this, a graph of mass against extension can be plotted, and a value for gravitational field strength calculated.

## Results

### Part 1

 Mass (kg) T (s) T^2 (s^2) 0.133 0.46905 0.22001 0.183 0.54889 0.30128 0.234 0.60876 0.37059 0.284 0.66865 0.44709 0.334 0.73851 0.54540 0.385 0.78840 0.62157 0.435 0.83830 0.70275 0.485 0.88821 0.78892 0.535 0.92813 0.86143 0.587 0.96805 0.93712

The graph of mass against period squared is shown below:

The spring constant of the spring can now be calculated in the same way as in experiment 1.

Uncertainties

Calculating Uncertainty in k

The LINEST function in excel is used to calculate the uncertainty in the gradient and therefore in k.

Using LINEST, the uncertainty in the gradient is:.

Therefore the value for k obtained is:

Uncertainty in Each Point

The uncertainty in the mass can be seen as ±0.001kg,as it is a digital balance so the reading error is ± half a scale division.

Conclusion

When measuring the period of the oscillations of the SHM in part one of the experiment, the spring would often swing back and forth, often leaving the path of the motion sensor, which caused the result to be ruined and the attempt for the mass had to be repeated. In order to overcome this, the spring was pulled down as straight as possible, to prevent this happening.

A CD was attached to the bottom of the spring, in order to increase the area that the motion sensor “sees”. This increased the accuracy of the motion sensor, as it is more likely to see a larger area, thus improving results. The mass of the CD was taken into account when weighing the masses.

It was found that the motion sensor worked best when closer to the spring. In order to facilitate this, a labjack was placed on top of a stool, to get the motion sensor as close to the spring as possible. The labjack also provided a level surface for the motion sensor, instead of the curved surface of the stool.

Overall, the experiment can be seen as a success with measurements being precise and accurate, giving a small error. This shone through in the results, where the value obtained for gravitational field strength was very close to the generally accepted value.

This student written piece of work is one of many that can be found in our AS and A Level Fields & Forces section.

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## Here's what a teacher thought of this essay

3 star(s)

Overall this is a good report which addresses many of the key points in a practical.

It could have been improved further by taking repeated readings and being more realistic about the uncertainty in T. 3 stars

Marked by teacher Pete Golton 06/06/2013

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