Investigating the Oscillations of a Mass on a Spring

Investigating the Oscillations of a Mass on a Spring

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Investigating the Oscillations of a Mass on a Spring

Aim:

In this physics coursework, I'm here to investigate the oscillations
of a mass of a spring. In this investigation, the oscillation means
the wave moving with periodic regularity. In this investigation, I can
use any mass and many springs, so that I can investigate the
oscillations.

Variables:

I believe there are many factors or variables, which can affect the
time for 1 oscillation. These can be:

· Mass of weight - I believe it will have a very big impact on the
time for oscillations.

· Number of springs - The number of springs will affect the affect the
time for oscillations a lot just like the number of mass, because of
the strength of the springs, and this depends on the number of
springs. The number of springs can affect the strength of springs and
this depends on the arrangement of the springs, which will be shown
much more detailed below.

· Arrangement of springs - First of all, there are 2 ways to arrange
the springs, and they are: Series or Parallel. Springs in series
extend further than springs in parallel. Also, during the trial
experiment I discovered that springs in parallel do not extend in a
straight line, they move from side to side and the springs can be
tangled up and this could be a major problem. Therefore, this would
affect the time taken to complete the given number of oscillations.
So, I will only do the springs in series, as the longer the extension,
the more accurate and complex the results will be. So, the arrangement
of springs will also affect the time taken to complete the given
number of oscillations. It can affect the spring constant, because
when the n number of springs of the same type is used in parallel, the
value of spring constant is n times larger than the spring constant of
one spring. When n springs of the same type are used in serial, the

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value of spring constant is 1/n of the spring constant of one spring.

· The efficiency of the spring - The springs keep on converting energy
between the forms of elastic potential energy and kinetic energy. But
the conversions cannot be 100% efficient, which means some energy is
lost in the form of heat energy (by air resistance or friction) or
sound. This is the main reason that the oscillation will eventually
stop. However if we always choose the same type of springs, their
efficiencies should be similar and the results would be much more
accurate.

· The different types of springs - The different types of springs
often have different spring constants, as they may be made of
different materials, or the thickness of the wire of the spring may be
different, or they may have different lengths etc.

· Air resistance - There is no way we can measure the air resistance,
because we will need very high tech equipment for this job. Although
some might think air resistance will affect only a little, still it
will affect the time taken for a certain number of oscillations.
However, I believe air resistance will have little effect on the wave
due to the small distance it oscillates. The effect of air resistance
could be unimportant, but the energy loss from potential energy to
heat energy could be from the air resistance.

· The time (dependant variable) -The output variables will be the ones
most affected by the mass on the spring.

· Amplitude (height) - I believe that the amplitude will affect the
time of oscillations, because the mass will speed up as the height
goes up, but I have not proven this yet, so I will do this first for
my investigation.

· Newton's second law - If there is a resultant force, the object
accelerates.

· Information given to us - Pull of load is larger than pull of spring
at the start and therefore accelerates. At the middle, the pull of the
load becomes equal to the pull of the spring (equilibrium), and
therefore the velocity is constant, and at its fastest - Newton's
first law. From the middle to the bottom, I believe the velocity will
decelerate.

Hypothesis (Prediction):

I will predict that the amplitude will affect the time of 1
oscillation. If the mass starts off at much higher position than the
normal position, then the time for 1 oscillation will be high. But if
the mass is let go at a bit higher than the normal position of the
mass, then the time for 1 oscillation will be higher. So basically, I
am saying that the amplitude will affect the time of oscillations in a
given time. As the amplitude goes up, the time will decrease. I think
the mass will speed up more if it is let go at higher amplitude. But
there is also a distance, which could also affect the time.

But I believe as the number of springs goes up and lined up in series,
the time for one oscillation will take longer. Because Newton's second
law states that F = M x A.

Therefore, acceleration is A = [IMAGE] which means acceleration is
inversely proportional to the mass, but in my case, I think the mass
is equal to the number of springs, because they both act the same way,
which is to affect the time of oscillation. We are only going to
consider 2 forces for simplicity, which are gravity and the force of
the spring. The only motion will be in the vertical direction, and it
will not be allowed to swing or rotate. For different masses making
the same oscillation, the forces are the same. Therefore, for bigger
masses or longer springs, the accelerations will be smaller and thus
the velocity is also smaller. It will take a longer time to complete
the same oscillation.

I also believe the maximum velocity will be in the centre of the
oscillation. This is because the resultant force changes direction
when the mass crosses equilibrium. Just before crossing the
equilibrium position, the mass is still accelerating. But just after
crossing equilibrium, the resultant force is directly opposite to the
velocity, therefore, it decelerates. So the maximum velocity will
occur in the middle of the oscillation.

I think that the extension of the spring will be proportional to the
load to a certain extent. I also believe that mass of the weight is
inversely proportional to the frequency. ([IMAGE]).

For the experiment, I am also doing on the acceleration and velocity
of the graph. I think that the spring will accelerate first and in the
middle, it will travel at a constant velocity and decelerate of the
other end. It will do the same going up or down and will give a bell
graph with time against velocity.

[IMAGE]

As you can see on the graph above, I believe, as the number of springs
gets higher, the time for one oscillation will take longer. And this
is a kind of graph that I expect to have in the end.

[IMAGE]

This is sort of a graph that I am expecting for the time against the
velocity. The spring will accelerate quickly and then in the middle,
it will travel at a constant velocity, and in the end the spring will
slow down.

Variables:

I first need to decide my variables in this investigation.

Input variable: The number of springs (the strength of springs)
instead of the mass. I feel the strength of the springs is much easier
to do, even though the mass is similar to the strength of the springs.

Output variable: I will be doing 2 dependent variables. Time for one
oscillation; in another words the period (frequency) will be my
output. I will do 10 oscillations then divide it by 10 at the end to
get the average of one oscillation. Also I will do the acceleration
and the velocity of the load for the output variable.

Information:

In order to do this investigation, I will need to know about the
oscillations deeply and clearly. Oscillation is repeated motion back
and forth past a central neutral position, or position of equilibrium.
A single motion from one extreme position to the other and back,
passing through the neutral position twice, is called a cycle. The
number of cycles per second, or hertz (Hz), is known as the frequency
of the oscillation. To find out the frequency, this formula can be
used:

[IMAGE]

[IMAGE]

[IMAGE][IMAGE]

A swinging pendulum eventually comes to rest if no further forces act
upon it. The force that causes it to stop oscillating is called
damping. Often the damping forces are frictional, but other damping
forces, such as electrical or magnetic ones, might affect an
oscillating spring.

Theory:

I have predicted that if the spring is in a vertical line, then the
amplitude of the mass will affect the time of 1 oscillation, because I
think the mass will speed up as the amplitude goes up. That's why the
amplitude or the height will affect the time or the period of
oscillations, but I will need to wait and see until I do the
experiments to prove this, and see whether I am right or wrong. I also
said that the extension of the spring would be proportional to the
load to a certain extent. This is called the elastic limit, where the
spring's strength doesn't seem to follow its graph's trend. Hooke's
law is the extension, which is proportional to the load. Also the mass
of the weight is inversely proportional to the frequency. ([IMAGE]) Or
([IMAGE]).

I said in my hypothesis that the amplitude does affect the time for
one oscillation, and I will try to prove that. Here are the results,
which were collected.

Results table to prove that the amplitude has an affect on the time of
oscillations:

Number of springs

Load (N)

Distance (cm)

Time taken to do 10 oscillations (s) 1st attempt

Time taken to do 10 oscillations (s)

2nd attempt

The average of 10 oscillations (s)

The average of 1 oscillation (s)

3

4

5

12.99

13.01

13

1.3

3

4

10

13.17

13.05

13.11

1.311

3

4

15

13.32

13.31

13.315

1.3315


As you can see above, most of the results for the time taken to do 10
oscillations are nearly the same, even though the amplitude was
different. This means that the amplitude doesn't matter when the load
is let go and start oscillating. So the height won't affect the time
of oscillations, so when letting the load go off the spring, it
doesn't matter where you let go of it, because it won't affect the
time of oscillations. So from now on, I will not measure the height
(distance) anymore for each experiment. So by looking at the results,
my hypothesis about the amplitude was wrong, because I said that the
amplitude does affect the time of oscillation, but actually, it
doesn't.

The prediction that is stated above is based on many experiments
carried out previously, including the trial experiment for this
investigation to find out whether the amplitude affects the time of
oscillations. It is also supported by various theories and laws, which
are explained below. First of all, it is common sense that the as more
springs are used (series), the further down it will get pulled so the
amplitude of the extension depends on the force. I have also stated
the extension is proportional to the load, unless too much load is
used which will go pass the elastic limit.

Firstly, this prediction is from experiments with load and its effect
on the distance (length) of 1 spring when it extends. We have found
out that the extension was proportional to the load 'E µ L' to a
certain extent. The certain extent is called the elastic limit. The
beginning of the extension up to the elastic limit is called the
Hooke's Law. This is very important in this investigation.

Once the elasticity limit of the spring has been passed, it loses its
functions and ignores Hooke's Law. In the beginning, the spring is
very strong so the extension is very short, but as it passes the
elastic limit point, the spring's extension gets longer and longer.
Therefore I must be careful with this and must not put too much load
onto the spring. It must be just the right and enough amount so that
it won't cross the elastic limit point.

Therefore, I believe the same pattern could be followed in my
investigation, because I am using springs and the mass, where the
current input variable in my experiment is the number of springs.

My prediction is made relating to the velocity in different parts of
the oscillation which is based on Newton's first and second laws. As I
am investigating half an oscillation using a ticker-tape-timer, and I
believe that at the 2 peaks, it will be at its slowest velocity and
fastest at the centre of the oscillation (equilibrium). This is
because firstly, starting at the top of the oscillation, pull of load
is larger than pull of spring in the beginning and therefore it
accelerates. In the middle, where there's equilibrium, the pull of the
load becomes equal to the pull of the spring so therefore the velocity
is constant, and at its fastest - Newton's first law states that if
there is no resultant force, the object will remain stationery or
maintain constant velocity and as you can see, the velocity is
constant in this case. From the middle to the bottom, I believe the
velocity will decelerate due to the air resistance and potential
energy turning into heat energy from the friction where the mass is
collided with the air molecules. However, I believe this will not have
a very significant impact on the results. But as the time passes, the
oscillation will take place very slowly gradually, but in the end, it
will suddenly go stop, due to the air resistance and the potential
energy turning into heat energy from the friction where the mass
(load) collide with the air molecules.

The equilibrium state:

[IMAGE]

If Resultant force = Downward force

Then this state is called 'Equilibrium'




First of all by looking at the diagram, the resultant force is equal
to the downward force and the mass and the spring remain unmoved due
to the Newton's first law, and we call this equilibrium, where the 2
forces are equal. Now, when the spring is stretched by an external
force, the weight moves down from its balanced position. There is now
a displacement of the weight from its balanced position. And it is
greater than the downward force (gravity) and the upward force
(resultant force). Therefore, the spring and the weight accelerate
upwards/downwards and the oscillation begins. The acceleration can be
calculated with Newton's Second Law:

[IMAGE] Or [IMAGE]

But in my case, the acceleration can be defined as this: [IMAGE] If we
used a larger mass, the same forces would be seen, but lower down.

As you can see, it is now a common sense that as the number of springs
gets higher, the lower the frequency (number of oscillation in a given
time), and as the number of springs gets fewer, the higher the
frequency.

We can also see the oscillation in terms of energy. The total amount
of elastic potential energy and kinetic energy is constant; therefore
there is always the transformation between the two forms of energy
when the spring is oscillating. Therefore it has the basic and
universal characteristics of all waves, which means the transfer of
energy.

Fairness Precautions during the experiment:

There are a few things that I will need to carry out in this
experiment to keep it as fair as possible to get the most accurate
results. I will use the same amount of load for every experiment to be
fair. As soon as the load comes up and is about to go down again, I
will then start my stopwatch. I will try to use springs, which all of
them have the same length to start off with. And when every experiment
has finished, I will put on another spring in series. This will mean
that the extension will be proportional to the load. Also, I will
record the time to the nearest 2 decimal places.

1st experiment apparatus List:

1. 3 Springs

2. Stand and Clamp

3. 4N of Mass (Load)

4. Stopwatch

5. A ruler

Diagram of the Apparatus:

To prove that the amplitude does not affect the time of oscillations,
I used this apparatus as shown below:

[IMAGE]


In this first experiment, I had to find out whether the amplitude does
affect the time of oscillations or not. So, I first collected the
apparatus as shown above, and then I used 3 springs + 4N of mass for
each experiments, which were 3 in total. The only differences in these
3 experiments were that they all were dropped at a different height to
see whether the height affected the time of oscillations or not. I did
the experiment twice for each 3 experiments so that I could take the
average which then the results would be far more accurate. The results
all came out nearly the same, which then I concluded that the
amplitude had no affect on the time taken.

By using this apparatus, I was able to see whether the amplitude
affects the time of oscillations. The results for this experiment was
already shown above, and the results showed very similar results even
though all of them started at different heights.

Procedure:

1. Set up the apparatus as shown above

2. Collect 3 springs and 4N of mass

3. Use the ruler to measure the height of the mass and pull it down
to a certain distance (5cm), so that when you let go, it will
start oscillating

4. Start the stopwatch as soon as letting go of the mass

5. Time how long it takes for the mass to do 10 oscillations

6. Repeat the whole procedure again with the same distance (5cm), so
that the average can be taken from the 2 results

7. Repeat the whole procedure again, but with only different
amplitude (10cm)

8. Repeat this procedure again with the same amplitude as the last
one (10cm)

9. Repeat the whole procedure again, but with only different
amplitude (15cm)

10. Repeat this procedure again with the same amplitude as the last
one (15cm)

11. Compare all the results between all three different experiments

2nd experiment apparatus List:

1. Springs

2. Stand and Clamp

3. Weights

4. Power supply

5. Ticker tape timer (Including carbon paper)

6. Ticker tape

7. Cello tape

Diagram of the Apparatus:

This experiment was to see whether the number of springs affected the
acceleration and velocity of the oscillations.

[IMAGE]

Oscillation Direction



[IMAGE]


Procedure:

To measure the difference in acceleration in an oscillation of a
spring, I set up the apparatus as shown in the diagram above. Then we
pulled down the springs with the weights on and attached it to the
bottom of the weights and pulled it through the ticker tape timer, and
then we turned on the timer and let go of the spring. We had to be
careful not to let the spring start to go down again, because the tape
would then have extra dots on, which could ruin our results. We did
this 4 times for the spring going upwards and repeated the whole
procedure for the spring going downwards by just putting the ticker
tape timer on top of the stand, and attaching the tape to the top of
the load.

3rd experiment apparatus List:

1. Springs (Up to 7 springs)

2. Stand and Clamp

3. 4N of Mass (Load)

4. Stopwatch

Diagram of the Apparatus:

[IMAGE][IMAGE][IMAGE][IMAGE] This experiment was to see whether the
number of springs actually affected the time of oscillations.

In this first experiment, I had to find out whether the amplitude does
affect the time of oscillations or not. So, I first collected the
apparatus as shown above, and then I used 3 springs + 4N of mass for
each experiments, which were 3 in total. The only differences in these
3 experiments were that they all were dropped at a different height to
see whether the height affected the time of oscillations or not. I did
the experiment twice for each 3 experiments so that I could take the
average which then the results would be far more accurate. The results
all came out nearly the same, which then I concluded that the
amplitude had no affect on the time taken.

By using this apparatus, I was able to see whether the amplitude
affects the time of oscillations. The results for this experiment was
already shown above, and the results showed very similar results even
though all of them started at different heights.

Procedure:

1. Set up the apparatus as shown above

2. Collect all 7 springs and 4N of mass

3. Using only 1 spring, attach 4N of mass onto the end of the spring
and pull it down a bit to start the oscillations. It doesn't
matter how much you pull the mass down, because previously, I have
proven that the amplitude does not affect the time of
oscillations, but it is advised to pull it down just a little, so
it is much easier to count the oscillations

4. Start timing as soon as letting go of the mass. Count up to 10
oscillations and stop immediately when it finishes. (I did this
with my partner and we both did it twice, which sums up to 4
results which were collected for each experiment)

5. Repeat the whole procedure until 7 springs are used

6. Look at the results and analyze


Data Presentation
-----------------

After the all the experiment, results would have been collected. The
average of the time of 10 oscillations will be calculated. The
frequency of the oscillation will be calculated from the time period.
These figures will be put into the table.

Expected Results:

I will be expecting some graphs plotted from the results, which will
be collected from the experiments.

I believe there is a connection between the mass and the length
(number) of the springs, and the relationships between them will be
something like this:

[IMAGE]

A straight line is expected when drawing the graph of load against
length of spring. Normally the graph should be drawn with the length
of spring against the load. The y-intercept of the regression line is
expected to be less than zero, because when there is no load on the
spring, the spring should still have a length.

[IMAGE]

As the number of spring increases, it is common sense that the time
(period) also increases as well direct proportionally. Therefore, I
expect this kind of results to be collected, where as the number of
spring increases, the time (period) increases as well.

[IMAGE]

As you can see above, the frequency should be inversely proportional
to the mass, and therefore, the graph should come out like this. This
is a typical graph of an inversely proportional graph, where as the
mass increases, the frequency gets smaller and smaller.

[IMAGE]

But if we plot this points onto number of springs against the 1 /
frequency, then the graph should come out as a direct proportional
straight linear line graph, just like the one above, where the
straight linear line goes through the origin (0,0).

Results table to see whether the number of springs have an affect on
the time of oscillations:

Number of springs

Load (N)

Distance (cm)

Time taken to do 10 oscillations (s) 1st attempt

Time taken to do 10 oscillations (s) 2nd attempt

Time taken to do 10 oscillations (s) 3rd attempt

Time taken to do 10 oscillations (s) 4th attempt

The average of 10 oscillations (s) 1st + 2nd+ 3rd+4th attempts

The average of 1 oscillation (s)

1

4

-

8.28

8.35

8.31

8.32

8.315

0.8315

2

4

-

10.66

10.81

10.89

10.65

10.7525

1.07525

3

4

-

14.25

14.26

14.19

14.32

14.255

1.4255

4

4

-

14.70

15.06

15.10

15.26

15.03

1.503

5

4

-

20.44

20.53

20.31

20.48

20.44

2.044

6

4

-

18.47

18.74

18.62

18.76

18.6475

1.86475

7

4

-

22.18

22.12

22.34

22.22

22.215

2.2215

The actual masses of the weights are in Newtons. However, I can change
the units to kilograms by using the formula F = M x G.

For example, using the weight as 1 Newtons, we could substitute this
into the formula to give:

1 = M x 10N/Kg

M = 0.1Kg

Graph for the results above:

[IMAGE]On the graph above, I took the average time of 10 oscillations
from 4 attempts, and this is how it looks. The trend on the graph
looks just as what I had expected, because as the number of springs
increases, the time it took to do 10 oscillations increases, more or
less direct proportionally. This graph tells me that my hypothesis
about the number of springs against the period per second was correct.

Ticker tape timer results (When moving upwards):

Number

Time (s)

Velocity (cm/s)

1

0.02

11.7

2

0.04

15

3

0.06

20

4

0.08

26.7

5

0.1

33.3

6

0.12

40

7

0.14

48.3

8

0.16

58.3

9

0.18

68.3

10

0.2

78.3

11

0.22

85

12

0.24

90

13

0.26

95

14

0.28

98.3

15

0.3

103.3

16

0.32

116.7

17

0.34

120

18

0.36

123.3

19

0.38

123.3

20

0.4

126.7

21

0.42

125

22

0.44

125

23

0.46

126.7

24

0.48

131.7

25

0.5

125

26

0.52

121.7

27

0.54

120

28

0.56

118.3

29

0.58

111.7

30

0.6

106.7

31

0.62

103.3

32

0.64

98.3

33

0.66

90

34

0.68

86.7

35

0.7

81.7

36

0.72

73.3

37

0.74

65

38

0.76

53.3

39

0.78

48.3

40

0.8

40

41

0.82

31.7

42

0.84

23.3

43

0.86

16.7

44

0.88

10

[IMAGE]

As you can see on the graph, the line is moving upwards, and the
points (results) are very good. This is because the trend is going
through most of the points, which means I can conclude that the time
of oscillations has a very good relationship with the velocity of
oscillations. But there are some points on the graph that which do not
follow the rules like the others. This means the ticker tape timer is
not 100% efficient, and only half of the graph is shown, which is not
very good to us. And I will do another same experiment to see the
other half of the trend. The greatest velocity on this graph can be
found out. It starts at 0 and finishes at 0.9, which means the middle
point is 0.45. And if we go along the y-axis, then it comes to around
125cm/s, which is the greatest speed above all the other ones.

Ticker tape timer results (When moving upwards):

Number

Time (s)

Velocity (cm/s)

1

0.02

-8.3

2

0.04

-13.3

3

0.06

-20

4

0.08

-30

5

0.1

-38.3

6

0.12

-46.7

7

0.14

-56.7

8

0.16

-63.3

9

0.18

-71.7

10

0.2

-81.7

11

0.22

-90

12

0.24

-101.7

13

0.26

-108.3

14

0.28

-118.3

15

0.3

-126.7

16

0.32

-133.3

17

0.34

-141.7

18

0.36

-146.7

19

0.38

-151.7

20

0.4

-156.7

21

0.42

-161.7

22

0.44

-161.7

23

0.46

-160

24

0.48

-163.3

25

0.5

-163.3

26

0.52

-161.7

27

0.54

-158.3

28

0.56

-156.7

29

0.58

-155

30

0.6

-146.7

31

0.62

-136.7

32

0.64

-126.7

33

0.66

-115

34

0.68

-103.3

35

0.7

-90

36

0.72

-78.3

37

0.74

-65

38

0.76

-53.3

39

0.78

-45

40

0.8

-35

41

0.82

-28.3

42

0.84

-21.7

43

0.86

-15

44

0.88

-10

[IMAGE]

As you can see above on the graph, this shows the other half of the
graph, which is going downwards. On this graph, the greatest velocity
on this graph can be found out. It starts at 0 and finishes at 0.9,
which means the middle point is 0.45. And if we go along the y-axis,
then it comes to around -165cm/s, which is the greatest speed above
all the other ones. Even though it is in a negative number, it doesn't
matte and is till the greatest velocity. The trend also does not go
through all of the points, which means the ticker tape experiment is
not very accurate. As the ticker tape passes through the timer, the
dots are printed on the tape. This means load briefly stops and also
the friction is created, which can be the cause of the abnormal
results. Therefore I must think of another experiment which can show
the full trend (going up and down) and which also does the experiment
very accurately.

By looking at both graphs, I can suggest that the mass is travelling
at its greatest velocity in the centre of the oscillation. Looking at
the graph, it is similar to a sine wave. I know that the greatest
velocity is in the middle, because both graphs above show the similar
results, which was the maximum velocity was found at the centre of the
oscillations.

Evaluation:

From the results we obtained and the graphs that have been analysed, I
can evaluate both investigations that the velocity and the period were
both successful. In this experiment, all of my predictions are proved
to be correct and there were no anomalous results. There were a few
points on the graph, which is out of the pattern. However, the
investigation went very well. So, there are still things that can be
improved and I still had problems with, which could have affected my
results.

* In my theory, I said that there were frictions and air resistance,
which would slow down the oscillations, and all the springs
wouldn't be 100% efficient. So according to that, the oscillations
would finally come to an end.

* Hooke's Law is only works under certain conditions. If the force
applied to a spring were beyond the elastic limit, the spring
would no longer obey Hooke's Law, which can cause inaccurate
results.

* In the beginning, the amplitude was a variable, but later I found
out that the amplitude could not be a variable, because of the
experiment that I did which proved that the amplitude did not
affect the time of oscillations. But if the amplitude is very big,
because of the heavy mass, the spring may be stretched so much
that they no longer would obey Hooke's Law. But this did not
happen to my experiments, because it was done very carefully, so
that I would not go over the elastic limit. So in the end, the
amplitude was not thought of a variable.

* It was impossible to make the springs oscillate 100% vertically.
The springs were always waving about, which would have caused
inaccurate and abnormal results.

* I would say that the timing was very inaccurate, even though it
was done carefully and done as best as I could. It was hard to
judge exactly by eyes where one period ends and where the next one
starts, and also the slow reactions to click the timer with the
human muscles.

* I did not weight the weights myself. But the actual mass of them
could be slightly different from what they were labelled, because
those were quite old and were used for many years, which could
have changed the results.

* It was hard to select two springs, which were the same. It was
nearly impossible to select 7 springs which ought to be the
exactly the same, but the best were selected in order to get the
best results.

For the ticker tape timer experiments, there are a few things we can
do to improve it as we discovered that dropping the mass did not prove
as good results as when we pulled the mass down and released it. We
must think of another method, which can improve the quality of
results. This would give a much better curve (trend) with less parts
missing. I have conducted a new procedure, using a magnetic sensor and
a computer. The set-up and data are shown below, using 2 springs and
2N's of mass. Here are the results, which were collected with this
method.

Diagram of the Apparatus:

[IMAGE]


Results:

Voltage (mT)

Time (s)

0

-6.1

0.05

-6.7

0.1

-7.2

0.15

-7.4

0.2

-7.3

0.25

-6.9

0.3

-6.2

0.35

-5.4

0.4

-4.6

0.45

-4

0.5

-3.5

0.55

-3.1

0.6

-2.9

0.65

-2.7

0.7

-2.7

0.75

-2.8

0.8

-2.9

0.85

-3.1

0.9

-3.5

0.95

-4.1

1

-4.8

1.05

-5.6

1.1

-6.4

1.15

-7.1

1.2

-7.5

1.25

-7.5

1.3

-7.1

1.35

-6.4

1.4

-5.6

1.45

-4.8

1.5

-4.1

1.55

-3.5

1.6

-3.2

1.65

-2.9

1.7

-2.7

1.75

-2.7

1.8

-2.7

1.85

-2.8

1.9

-3.1

1.95

-3.4

2

-3.9

2.05

-4.6

2.1

-5.4

2.15

-6.2

2.2

-7

2.25

-7.5

2.3

-7.6

2.35

-7.3

2.4

-6.7

2.45

-5.9

2.5

-5

2.55

-4.3

2.6

-3.7

2.65

-3.3

2.7

-2.9

2.75

-2.8

2.8

-2.7

2.85

-2.7

2.9

-2.8

2.95

-3

3

-3.3

3.05

-3.8

3.1

-4.5

3.15

-5.1

3.2

-5.9

3.25

-6.7

3.3

-7.3

3.35

-7.5

3.4

-7.2

3.45

-6.7

3.5

-6

3.55

-5.2

3.6

-4.4

3.65

-3.9

3.7

-3.4

3.75

-3.1

3.8

-2.8

3.85

-2.7

3.9

-2.7

3.95

-2.8

4

-3

4.05

-3.3

4.1

-3.7

4.15

-4.3

4.2

-5

4.25

-5.8

4.3

-6.5

4.35

-7.1

4.4

-7.4

4.45

-7.2

4.5

-6.7

4.55

-6

4.6

-5.3

4.65

-4.6

4.7

-3.9

4.75

-3.4

4.8

-3.1

4.85

-2.8

4.9

-2.8

4.95

-2.8

0

-6.1

0.05

-6.7

0.1

-7.2

0.15

-7.4

0.2

-7.3

0.25

-6.9

0.3

-6.2

0.35

-5.4

0.4

-4.6

0.45

-4

0.5

-3.5

0.55

-3.1

0.6

-2.9

0.65

-2.7

0.7

-2.7

0.75

-2.8

0.8

-2.9

0.85

-3.1

0.9

-3.5

0.95

-4.1

1

-4.8

1.05

-5.6

1.1

-6.4

1.15

-7.1

1.2

-7.5

1.25

-7.5

1.3

-7.1

1.35

-6.4

1.4

-5.6

1.45

-4.8

1.5

-4.1

1.55

-3.5

1.6

-3.2

1.65

-2.9

1.7

-2.7

1.75

-2.7

1.8

-2.7

1.85

-2.8

1.9

-3.1

1.95

-3.4



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