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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
jumping-off-bridges 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 top-pan 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 retort-stand 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 straight-line 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 Y-axis
and mass (in kg) on the X-axis, 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 = kgT-2
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 kgT-2

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 kgT-2, and
using the calculation method I got an average elastic constant of
20.90 kgT-2.

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 best-fit 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 co-efficient 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 kgT-2

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 kgT-2 plus/minus 0.24,or, it has an
error range of 1.16%

Having caught the potential error of mis-stated 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.

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MLA Citation:
"Investigating the Vertical Oscillations of a Loaded Spring." 123HelpMe.com. 20 Apr 2014
    <http://www.123HelpMe.com/view.asp?id=120343>.




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