In this experiment, the spring is loaded with different masses of known values and the extension of the spring from the equilibrium position when there is no mass on the spring is recorded. Then a graph of extension against mass can be created from this data. The equation of the best fit line of this data can now be compared to and a value for the Earth’s gravitational field strength can now be calculated using this data and the value for the spring constant calculated in the first part of this experiment.
Procedure
Apparatus
- Analogue stopwatch, accurate to 0.1s
- Stand
- Two different springs
- Slotted masses and holder
- Reading disk
- Ruler
Part 1
- The holder and reading disk were attached to the spring.
- A 0.01kg mass was then added to the mass holder. The ruler was then adjusted so that the top edge of the reading disk was exactly aligned to the 0 mark on the ruler.
- The mass was then extended 0.04m downwards, and released. At this exact point, the stopwatch was started.
- After the mass had undergone 10 oscillations, the stopwatch was stopped, and this time for 10 oscillations was recorded. This was repeated 4 times for each mass, and an average was taken, in order to gain the most accurate results.
- This was then repeated for differing masses, in this case the masses were 0.01kg up to 0.1kg in 0.01kg increments. This part of the experiment was then repeated for the two different springs.
- It was found that the second spring could not take mass over 0.05kg, as it would hit the table when it was displaced the 0.04m for it’s equilibrium position. Instead, the mass on this spring went from 0.01kg to 0.04kg in 0.01kg increments.
- Graphs were then created of period squared against mass for each spring, and the spring constant for each spring could be calculated as previously shown.
Part 2
- In the second part, the ruler was first adjusted so that the top of the reading disk was exactly aligned to the 0 mark on the ruler.
- A 0.005kg mass was then added to the mass holder, and the distance the spring extended was recorded. Another 0.005kg mass was added, to make the total mass 0.001kg, was added and the distance the spring had extended from the 0 position was recorded again.
- This process was continued, increasing in 0.005kg increments, until the total mass on the spring was 0.1kg.
- This was repeated for each spring. Again, the second spring could not take as much mass as the first without hitting the table. Instead, the masses placed on it increased in 0.005kg increments but only up to 0.06kg. It was not deemed necessary to repeat the experiment for each spring as there is likely to be little error in this experiment.
- A graph of extension against mass could then be created for each spring and, using the value for the spring constant calculated before, a value for gravitational field strength could then be calculated as shown before.
Results
Spring 1 – Part 1
Taking the average time for 10 oscillations, calculating the period and the period squared:
So the graph of period squared against mass is:
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.
The equation for this graph can now be compared to to gain a value for the spring constant of this spring:
Since is the y value of the graph, and m is the x value of the graph, it follows that:
The spring constant, k, can be found by substituting the value of the gradient of the graph into this equation:
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:
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)
This uncertainty will be the same for k, because the other component’s involved it it’s calculation are all constants.
Therefore:
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 equation, 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
As a percentage of 2.9, this is 1.1%.
This is doubled, to 2.2% uncertainty in period squared, so:
All uncertainty calculations were done as shown above, and the results were:
Mass = 0.02kg
Mass = 0.03kg
Mass = 0.04kg
Mass = 0.05kg
Mass = 0.06kg
Mass = 0.07kg
Mass = 0.08kg
Mass = 0.09kg
Mass = 0.10kg
Because all these uncertainties are very small, the error bars on the graph are too small to see.
Spring 1 – Part 2
So the graph of mass against extension is:
The equation of this best fit line can now be compared to the equation .
Rearranging:
Since x corresponds to the y-axis of the graph, and m corresponds to the x vale of the graph, it follows that:
Substituting the gradient of the graph and the value for the spring constant of the spring already calculated, a value for g can be calculated.
Uncertainties
Calculating Uncertainty in g
Because the points do not deviate enough from the trendline to draw a parallelogram around it, the LINEST function of excel must be used to calculate the uncertainty in the gradient of the trendline in the mass against extension graph.
Using LINEST, the value for the gradient is:.
In calculating g, the values for k and the gradient of the graph are used, and these can be combined as shown below.
Therefore the calculated value for g in this experiment is:
Spring 2 – Part 1
So the graph of period squared against mass is:
From this, the equation shown for spring one can be used to calculate the spring constant for this spring:
Uncertainties
Calculating the Uncertainty for k
Because there are so few points on this graph, a parallelogram could not be drawn around the points, so the LINEST function of excel was used.
Using LINEST, the error in the gradient is
Thus the error in k is also 0.87, so:
Calculating Uncertainty for Each Point
To calculate uncertainty for each point, the same method as demonstrated for spring 1 is used.
Mass = 0.01kg
Mass = 0.02kg
Mass = 0.03kg
Mass = 0.04kg
Because the uncertainty in each point is so small, the error bars are too small to see on the graph.
Spring 2 – Part 2
So the graph of extension against mass is:
In the same was as for spring one, a value for g can be calculated from this graph and the spring constant calculated previously:
Uncertainties
Calculating Uncertainty in g
To calculate the uncertainty in g, the same method used for spring 1 is used.
Using LINEST, the value for the gradient is:
This is combined for the uncertainty in g.
So the value for g is:
Discussion
Conclusion
This experiment has shown the relationship for a mass m on a spiral spring with a period of oscillation T. Values for gravitational field strength were also calculated at and .
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.
