Investigating the Vertical Oscillations of a Loaded Spring
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Investigating the Vertical Oscillations of a Loaded Spring
Aims: The aim of this investigation is to find the elastic constant of the spring under study. The elastic constant of material is always useful to know and in some cases vital. For instance the elastic constant of a bungee rope is vital. It will tell the operator of the ride by how much the bungee rope will extend, using Hookes law (F=kExt.), and so how far the rider will fall. Using this they can calibrate the ride so as the give the rider the best possible experience. So a way of finding out the elastic constant of a material is important in today's jumpingoffbridges kind of world. Because of this the aim of this experiment is to find out through experimentation the elastic constant of a material and find out if it is the same as what is stated in the specifications of the material. The object under investigation is a metal spring. Method: To find the elastic constant of the spring I will attach a mass (a weight with its mass measured on the scales) to it, then stretch it to a set amount from its normal location (this being where the attachment hook rests when no other forces are applied). Doing this I will then let go and time the oscillations of the spring (one oscillation meaning leaving the starting position, passing the rest position, reaching the aphelion, then returning to the starting position, or simpler one cycle). To get an accurate measurement of period of the oscillations I will time how long it takes to reach 10 oscillations and then divide that by 10. I will do this 3 times for each mass and average the results so as to reveal and eliminate any possible errors. I will use five different masses (100g, 200g, 300g, 400g, 500g,) so as to get a full spectrum of results and ensure that I check that elastic constant does not change with mass (it should not as it is the constant of that material). Once having recorded the masses and their corresponding time periods I shall work out the elastic constant of the material in two ways, by calculation and by graph. Variables kept Constant: Ø Spring used shall remain the same. Ø Measuring equipment used shall be the same Ø Height of the retort stand shall be the same to combat difference in observations and keep gravity constant (though in theory this should not affect the time period). Ø Observer making the readings shall remain the same so as to cut down on human error. Variables Varied: Ø Mass on the end of the spring. I shall use the masses 100g, 200g, 300g, 400g, and 500g. Details of measurements taken: Ø I shall measure the time period for 10 oscillations and then divide this by 10 to get time period per oscillation. This should hopefully improve accuracy. Ø 5 different masses recorded, repeating each mass three times to get an average, again to improve accuracy. Apparatus: Ø Retort Stand and accompanying clamps. Ø The spring under investigation. Ø A series of weights each weighing 100g and their weight/hook hybrid counterpart allowing attachment to the spring. Ø A stopwatch or other form of accurate timer. Ø Digital Scales to measure mass (I will be using digital scales as the give a more accurate reading than a toppan balance). Diagram: [IMAGE] Safety: As I am only timing the time period of oscillations of a spring there are only a few safety measures I need to take into account. Firstly as I am using weights I will place the retort stand no where near the edge of the workbench so if the spring breaks, it minimises the chance of the weights striking my foot or other such weak object near the floor. I will also wear safety goggles, as I will be placing strain on the metal spring and if it breaks its whiplash could be very damaging. I will make sure that the clamp holding the spring and weights are over the base of the retort so as to insure that the retortstand will not fall over. Theory: In this investigation I shall be using the equation T2=4P2 x M K Where T is time period for one oscillation, M is mass used, and K is elastic constant. Analysis: As I said above I shall use two separate methods to work out elastic constant. Both involve the above equation, T2=((4P2)/K)x M. The calculation method involves rearranging that equation to give K as such T2 = 4P2 x M = K = 4P2 x M K T2 The Graphical method means reading the first equation as a standard Y=mX+C straightline equation. T2 = 4P2 x M K Y = mX+C Where Y= T2 , m=4P2/K and X=M, while C=0. I can draw a graph with Time period squared (in seconds) on the Yaxis and mass (in kg) on the Xaxis, and then plot my results. With that done I can then draw a line of best fit and work out its gradient using rise over tread. Knowing the gradient (m) I can work out K as such: m = 4P2 K Which is the same as: K= 4P2 m So the elastic constant of the spring equals ((4P2)/ T2)x M and 4P2/gradient of plotted graph. As K is equal to K = 4P2 x M T2 We can say the units of K are equal to the units of M divided by the units of T2. So the units of K are kilograms divided by Seconds squared or rather Units of K = kgT2 Results Table: Used Mass (kg) Measured Mass (kg) Time period for 10 Oscillations (s) 1 Oscillation = T (s) T2 (s) 0.1 1st 0.993 4.38 2nd 4.56 3rd 4.44 0.1 average 4.46 0.45 0.20 0.2 1st 0.197 6.03 2nd 6.10 3rd 5.94 0.2 average 6.02 0.60 0.36 0.3 1st 0.296 7.38 2nd 7.57 3rd 7.45 0.3 average 7.47 0.75 0.56 0.4 1st 0.393 8.37 2nd 8.50 3rd 8.38 0.4 average 8.42 0.84 0.71 0.5 1st 0.491 9.34 2nd 9.43 3rd 9.25 0.5 average 9.34 0.93 0.87 Analysis: From the graph we can see that the gradient is equal to 0.5199 meaning that 4P2/K = 1.8104. Therefore K (the elastic constant) is equal to 4P2/1.8104 or rather 21.81 kgT2 Using the calculation method I shall use the first third and fifth masses. T2 = 4P2 x M = K = 4P2 x M K T2 1st reading: K = 4P2 x 0.099 0.20 K = 19.54 2nd reading: K = 4P2 x 0.296 0.56 K = 20.87 3rd reading: K = 4P2 x 0.491 0.87 K = 22.28 As you can see the two recorded results are not that different. Using the Graph method I worked out an elastic constant of 21.81 kgT2, and using the calculation method I got an average elastic constant of 20.90 kgT2. The results I got, complementing each other, do suggest a high degree of accuracy. From the graph I plotted we can see that there is a quantitative correlation between the mass on the spring and its time period for oscillations. This fits well with the theory that a constant, or rather the elastic constant, govern the time period of oscillations for a material. All the results I worked out fitted with the line of bestfit indicating that there are no anomalous results. Like wise the similarity of the values of K obtained from the calculation method indicates that there is a definite ink between mass and time period squared and this indeed is a coefficient of the elastic constant. And the similarity between the results from both methods indicates that we have accurate results. I need to find out the actual elastic constant of the spring I used to find out which, if either, is the most accurate method for working out the elastic constant of a material. Once that is found I can decide which method is best at working out the elastic constant. Error analysis: As in most experiments human error will have been the biggest offset. Things like judging when 10 cycles are finished and the reaction time in starting and stopping the timer will add to the inaccuracy of the experiement. A large error I avoided was not taking the masses as what they were stated and measuring hem independently. As you can see above the masses were wildly different from their stated masses. At the end of this report you can see in a second graph how different the results would have been if I had used the stated masses. One error that I could not avoid was the time keeping accuracy. Not only was there human error (this was minimised by using the same human) but the stopwatch was only accurate to plus/minus 0.005 s. In a third graph you can see the potential error. A large error was avoided by confirming the graph was plotted correctly. Without this correction I would have had an error of roughly 340% (Plotting the axis the wrong way round gives the elastic constant as roughly 75 and (75/22)*100 = 340). If we drop the possible errors in to the equation we can work out the possible error in the elastic constant. Avoided Mass error: m = 1.78 K = 4P2/1.78 or rather 22.18 kgT2 So by measuring the mass I avoided an error of 0.37, or, an error of 1.6% Upper and lower range of Elastic constant: Upper Gradient = 1.8243 Lower Gradient = 1.7965 Upper K = 4P2/1.82 or rather 21.69 Lower K = 4P2/1.79 or rather 22.05 So the elastic constant is 21.81 kgT2 plus/minus 0.24,or, it has an error range of 1.16% Having caught the potential error of misstated masses I'm quite confident of the experiments accuracy. To improve this investigation I could rig up a more accurate timing device that used the oscillating load to start and stop the timer rather than just using a human observer. This would cut down on human error. As an extension to this investigation I would study many different springs to confirm that the method applies to all materials, and was not just a chance fluke. I could also study whether heat and other environmental constants affect the elastic constant of a material. Appendix contents: Second graph showing potential error of using stated masses rather than measuring them. Third Graph showing possible error derived from the stopwatch. In Lab results table used. How to Cite this Page
MLA Citation:
"Investigating the Vertical Oscillations of a Loaded Spring." 123HelpMe.com. 20 Apr 2014 <http://www.123HelpMe.com/view.asp?id=120343>. 
