Factors Affecting the Time Period for Oscillations in a Mass-spring System


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Factors Affecting the Time Period for Oscillations in a Mass-spring System

When a mass is attached to the end of a spring the downward force the
mass applies on the spring will cause the spring to extend. We know
from Hooke's law that the force exerted by the masses attached to the
spring will be proportional to the amount the spring extends. F = kx

When additional downward force is applied to the spring we can cause
additional tension in the spring which, when released, causes the
system to oscillate about a fixed equilibrium point. This is related
to the law of conservation of energy. The stain energy in the spring
is released as kinetic energy causing the mass to accelerate upwards.
The acceleration due to gravity acting in the opposite direction is
used as a restoring force which displaces the mass as far vertically
as the initial amplitude applied to the system and the process
continues.

A formula that can be used to relate mass applied to a spring system
and time period for oscillations of the system is

T = 2π√M/k

This tells us T2 is proportional to the mass

To test this relationship an experiment will have to be performed
where the time period for an oscillation of a spring system is related
to the mass applied to the end of the spring.


Variables that could affect T

Mass applied to spring; Preliminary experiments should be performed to
assess suitable sizes of masses and intervals between different masses
used in the experiment.

Spring constant; The spring constant will be useful to confirm the
relationship. A simple force - extension experiment should be
performed to get an accurate value for k which can be compared to the
value of k from the final experiment.

Amplitude; The amplitude of the oscillations should be kept constant.
Bear in mind the amplitude cannot be larger than the extension caused
by the smallest mass applied to the spring as this would not allow the
system to oscillate properly.

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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 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

Mass (kg)

Extension (m)

0.100

0.04

0.200

0.07

0.300

0.10

0.400

0.13

0.500

0.17

0.600

0.21

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 4Ï€2 / 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

Mass (kg)

Time for 20 oscillations (s)

Average time for 1 oscillation (s)

Average time for 1 oscillation squared (s2)

0.100

8.13

0.40

0.16

8.02

7.96

0.200

11.54

0.57

0.32

11.34

11.56

0.300

13.95

0.69

0.48

13.77

13.84

0.400

15.46

0.79

0.62

16.09

15.86

0.500

17.20

0.87

0.76

17.52

17.20

0.600

19.13

0.96

0.92

19.26

19.24

0.700

21.19

1.04

1.08

20.73

20.53

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.


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