Elastic limit of spring; If the spring has been extended past its elastic limit it will have become permanently deformed and will no longer obey Hooke’s law causing inaccuracies in the readings.
Preliminary Experiment
From preliminary experimentation I found that;
- 0.03m is a suitable amplitude to give appropriate sized oscillations with minimal disruption at lower masses.
- Due to a ±0.01s error in the stopwatch it is sensible to only take readings of 1< second to keep % error below 1%. This can be achieved by taking the time for 20 oscillations and using that data to find the time for a single oscillation.
- Intervals of 0.1kg between masses of 0.1kg and 0.7kg give a wide range of readings without putting excessive strain on the spring.
No particular concern to safety is required under these conditions. If larger masses were to be used safety goggles would need to be worn in case of the spring snapping.
Apparatus
0.100kg masses – The 7 100g masses I am planning to use in my experiment actually have an average mass of 0.098kg meaning there is a percentage error of around 2% in the masses I am using in my experiment. Although this is a larger error than would be preferable, if the same masses are used in both the force – extension experiment and the mass – oscillation experiment the values for the spring constant should still be comparable.
Stopwatch ±0.01s – From preliminary experimentation I know that the smallest time I will probably have to measure will be around 6-7 seconds. This means than the largest percentage error I should encounter in the timing will be around 0.17% which will be negligible in this experiment.
1m Ruler ±0.001m – The smallest measurement I will have to take will be the 0.03m amplitude of the oscillations. The percentage error in this measurement will be about 3.3% which is quite large but if the amplitude was changed so as to give a percentage error of 1% or below we would have to use an amplitude of at least 0.10m which would not be possible at masses under around 0.300kg.
Spring – If the spring has been permanently deformed prior to the experiment it will no longer obey Hooke’s law (F = kx) so we will not be able to use the calculated value for the spring constant from the
force – extension experiment.
The main factor that will effect the experiment will be human error. It is very difficult to define exactly when an oscillation starts and ends and almost impossible to actually get the system to oscillate perfectly vertically. Also factors such as reaction times will have a significant effect on the results from the experiment.
Force – Extension
To determine the spring constant of the spring been used, a force / extension graph can be plotted. Using Hooke’s law we can see that if force is plotted on the Y axis and extension is plotted on the X axis the gradient will be the spring constant. In this case Mass is been plotted on the Y axis so the spring constant will be the gradient multiplied by g (9.81ms-2).
Method
- Set up the spring so as it hangs securely from a point high above the work area.
- Attach a mass of 100g to the end of the spring and measure the amount which the spring is extended by.
- Repeat the process at 100g intervals up to 600g recording the extension at each mass.
Results
This gives a value of 28.0 Nm-1
for the spring
constant of the spring
Time for oscillation - Mass
The formula T = 2π √M/k can be rearranged to T2 = (4π2/k) M. When compared to y = mx + c we can tell that with T2 on the Y axis and M on the x axis, the gradient will be 4π2 / the spring constant. We can also tell how accurate our results our by checking that c = 0 (i.e. the graph fits comfortably through the origin).
Method
- Set up apparatus as shown in diagram, making sure the spring hangs totally vertical.
- Attach the first mass of 100g to the end of the spring and make sure the system is in equilibrium.
- Pull down on the mass to give the spring potential amplitude of 0.03m and release, simultaneously starting the stopwatch.
- Record the amount of time the system took to oscillate about its equilibrium point 20 times and stop timing.
- Repeat the process at intervals of 100g up to 700g performing the test 3 times at each mass for accuracy.
Diagram
Results
These results give a value of 25.6 Nm-1
for the spring constant of the spring.
Analysis / Evaluation
The spring constant of the spring was calculated as 28.0 Nm-1 from my mass – extension experiment. This can be taken as a reasonably reliable calculation as the experiment did not have much potential for human error, there were less variables to be considered and my graph confirms the relationship F = kx (Hooke’s law) as it is a straight line through the origin.
There are some slightly anomalous points on the mass – extension graph probably due to the 2% error in the masses used or due to the spring becoming slightly deformed during the experiment.
As predicted from the relationship T = 2π√M/k, as the mass applied to the system increases, the time for an oscillation increases exponentially. The relationship can be confirmed as the graph of T2 against M is a straight line through the origin.
The spring constant calculated from this graph was 25.6 Nm-1 which is comparable to the 28.0 Nm-1 from the first experiment. Taking 28.0 Nm-1 as the ‘actual value’ (as it was calculated from a more reliable experiment with a simpler relationship) we have a percentage error of 8.6% between the two values with no particularly apparent anomalous results.
The error in the apparatus only came to around 2.021% (2% of that been the inaccuracies in the masses used) so most of the error came from the practical difficulties in actually performing the experiment.
The experiment was probably not entirely suitable the main problem been trying to get the system to oscillate as vertically as possible. If the system oscillates at just 10 degrees off the vertical then only 98% of the amplitude actually acts on the vertical component of the motion.
The angle at which the amplitude is applied also has a larger effect as the acceleration due to gravity will not act parallel to the motion of the oscillations if the system is not oscillating perfectly vertically. This will cause the system to gradually oscillate further from the vertical disrupting the results even more by the end of all 20 oscillations.
Another problem with the system not oscillating vertically was that the system began to almost swing rather than oscillate making it very difficult to actually pin-point the exact moment the oscillation was completed.
This problem would be very difficult to overcome with the experiment been performed manually. If the system was set up in a perfectly vertical plastic tube then the tube was removed just before the amplitude was released we could have more accurate readings as the human eye cannot easily judge how close something is to been vertical. We would then however encounter problems with friction between the plastic tube and the masses. It also proves very difficult to remove the plastic tube without disrupting the amplitude of the oscillation.
