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The aim of this experiment was to find out if the time period of a vertical mass oscillating system is dependant on the spring constant (k) and mass (m) by:
T = pkqmr
To do this, I completed three experiments.
Experiment A involved measuring the extension of a single spring with varied mass (of between 0.05 and 0.7kg. This will allow me to work out a value of k for the spring which I can use throughout the analysis.
In experiment B the time period of 20 oscillations of a single spring was measured, with a varied mass between 0.2 and 0.7kg.
Experiment 3 involved measuring the time period of 20 oscillations with a fixed mass and varied spring constants. The different spring constants will be created by different arrangements of springs, all with the same spring constant.
Experiment A - Determining k
To find the value of k, one spring was used, and the extension measured as the mass was increased.
Hooke s Law states that the extension of a spring is directly proportional to the force applied, provided that the elastic limit of the spring is not exceeded . The formula for this is F=kx, where F=force applied (N), k = spring constant (Nm-1) and x=extension (in m)
This means my results should plot a straight line with and equation of the form y=mx+c, where m is the gradient and c is a constant. In this case, the value of c should be zero, as when there is no weight, there would be no extension.
The gradient of the line is 2.507±0.065. As this is the gradient for a mass extension graph, the gradient of the force extension graph would be 9.81(2.507±0.065)= 24.59±0.64.
This is the value for k I will use in my analysis.
Laws of Logs
The data for the next two experiments will be manipulated(kq) + ln(pmr)
Using Law 3, ln(kq) + ln(pmr) = ln(kq) + ln(pmr)
This gives the equation lnT = qln(k)+ln(pmr)
Comparing this to y=mx+c, the gradient is q, and the y intercept is ln(pmr).
As lnT against lnk is a straight line, the original expression lnT= ln(pkqmr ) is in the correct form, and the values q and ln(pmr) can be found.
The gradient of my graph is -0.468±0.019, which is the value of q.
Using the point (2.90, 0.18) (the point where the line of max gradient and min gradient cross) the y intercept of the line can be found.
0.18 - (-0.449x2.90) =cmin cmin= 1.482
0.18 - (-0.487x2.90)=cmax cmax = 1.592
ln(pmr)= 1.54±0.06
Finding p
To find the value of p, I will use the previously found values of k, q and r along with the known mass. I will sub these values into the equations ln(pkq)= 0.21±0.02 and ln(pmr)= 1.54±0.06, then rearrange to find the min and max value of p.
ln(pkq)= 0.21±0.02
pkq=e(0.21±0.02)
p(24.59±0.64)(-0.468±0.019)= e(0.21±0.02)
p=e(0.21±0.02)(24.59±0.64)(0.468±0.019)
pmin = 6.06
pmax = 5.03
ln(pmr)= 1.54±0.06
pmr=e(1.54±0.06)
p(0.700±0.007)(0.474±0.117)= e(1.54±0.06)
p=e(1.54±0.06)(0.700±0.007)(-0.474±0.117)
pmin = 5.39
pmax = 5.65
Using the biggest and smallest values of p, p=5.55±0.52.
My values for p,q and r
p=5.55±0.52
q = -0.468±0.019
r = 0.474±0.017
Research on T = pkqmr
The oscillations of the spring and mass can be proven to be simple harmonic motion.
When the mass is in equilibrium, the force on the mass due to the spring upwards equals the force on the mass due to gravity downwards.
mg= kx0
When the mass is pulled below the equilibrium position, the force on the mass due to gravity is less than the force on the mass due to the spring. This means the resultant force would be upwards.
F=ma=k(x0-x)-mg
Substituting kx0 with mg gives
ma = mg -kx-mg
ma = -kx
a = (-k/m)x
This proves that the oscillation is simple harmonic motion because the acceleration towards the rest point is negatively proportional the displacement from the rest point.
This means that w2 =k/m
w = (k/m)0.5
And as T=2p/w
T=2p(k/m)-0.5
T=2p(m/k)0.5
Using Physics by Robert Hutchings, this can be confirmed as T = 2pk-0.5m0.5.
This means: p=2p
q=-0.5
r=0.5
Units
To find the units of p, k and q I will use the correct version of the formula (taken from Physics by Robert Hutchings.
T = 2pk-0.5m0.5
As the values of p and q are powers, they can have no units.
To find the units of p, I will work out the units on both sides of the equation then rearrange to find the units of p. P will be used to represent the units of p.
s = P(Nm-1)-0.5(kg)0.5
s = P(kgm/N)0.5
s = P(kgm/kgms2)0.5
s = P(1/s2)0.5
s = P/s
s2 = P
This means that the units for p are s2.
Bibliography:
www.hookeslaw.com/hookeslaw.htm
Physics 1 - Cambridge University Press
http://www.projectalevel.co.uk/maths/logs.htm
http://www.ndt-ed.org/EducationResources/Math/Math-e.htm
http://en.wikipedia.org/wiki/Simple_harmonic_motion
Evaluation
I feel like the experiment went well, although my results do not perfectly match the theory.
Actual Values for p,q and r:
p=2p
q=-0.5
r=0.5
My values for p, q and r:
p=5.55±0.52
q = -0.468±0.019
r = 0.474±0.017
This means that none of the values I found included the actual values in their error range.
This could be for many reasons.
Throughout my analysis I have treated the spring as and ideal spring. An ideal spring has no mass and has no damping losses. The spring is not ideal as it has a mass and damping losses, so making this assumption may have lead to errors.
Experiment A
During this experiment, parallax error was eliminated by taking readings with eye level to the measuring point (bottom of the spring).
The ruler was clamped close to the weights so an accurate reading could be taken, but not close enough to snag on the spring/weights.
The largest errors were on the masses. These errors could have been reduced by using a single 50g mass (error 0.001kg) instead of five 10g masses (combined error 0.005kg). Another way to overcome this would be to use a digital balance. This would give measurements for all masses to ±0.0001kg, which is significantly lower than the errors I experienced.
On the graph produced by my results, the lines of best fit do not include the origin. This could be due to a systematic error, as all the results are higher than expected, or due to the spring being stiff when little force is applied.
Experiment B
There is one anomalous result on the graph for this experiment, although it is very close to being correct and it has a very small error on mass. An error this close to being correct could be due to being slightly too quick/slow with the stopwatch.
Parallax error was eliminated in this experiment by using fiducial markers. This ensured I was taking measurements when the mass passed the correct point.
In this experiment the errors for time and mass were of comparable size. The errors on the weights could have been reduced (as described earlier) and the errors on the timings could have been reduced by either timing more oscillations or using a light gate to time the oscillations, which would reduce the human error on the timings.
Experiment C
From the graph it can be seen that I have 8 anomalous results, but there is still an obvious trend and most of these anomalous points are very near the best fit lines. There are many things that could have caused these errors.
During this experiment I have assumed the spring constant for all springs used to equal my result from experiment A (24.59±0.64 Nm-1). Although the springs were designed to have the same spring constant, it may vary between the springs. One way to overcome this would be to work out the spring constant of every spring used, using the method used for experiment A.
Springs touching each other and snagging was a problem. To reduce this, dowel was used to separate the springs, but this dowel created other problems as well.
The dowel used to stop springs from tangling/snagging will have had an effect. To reduce this effect, the dowel should be weighed and then the mass of the dowel should be taken into account. Dowel above an arrangement of springs will have no effect. When the dowel hung diagonally, a slight alteration of the position of the springs would make the dowel level again.
Occasionally the system would start to move sideways as well as vertically. This does not effect the time period but makes it hard to take readings, and the time period would be affected if the spring/mass hit anything. To help with this, the amplitude was kept small and the mass was pulled vertically then released.
I used a mass of 0.7kg, with an error of 0.007g. This error could be reduced by weighing the mass with an electric balance which would give and error of 0.001kg.
The errors on the time period would be hard to reduce as the biggest part of them is due to human error. One way to do this would be to use a larger mass (to make the time period larger and therefore the error less significant) or to use a light gate to record results.
Parallax error was eliminated in this experiment by using fiducial markers. This ensured I was taking measurements when the mass passed the correct point.
Safety
To avoid springs snapping and weights dropping, the mass of weights was limited to 700g.
Counterweights were placed on the bottom of the stands to ensure that the stands didn t topple over.
The stands were placed away from the edge of the table so they could not be knocked off the table easily.
When the amplitude of oscillations of certain spring combinations (in experiment C) were very large, a point higher up on the spring was indicated and used to take measurements. This makes no difference to the time period , but is safer as you can be seen easier and if stands/masses fall off the table they would not hit your head.
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