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Investigating Factors That Affect the Period Time of a Simple Pendulum

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Investigating Factors That Affect the Period Time of a Simple Pendulum


Planning

Definitions: Oscillation : Repeated motion of pendulum (to and for)
Period (T) : Time taken for one full oscillation

In this investigation, I am going to experimentally determine a factor
which
will affect the period of a simple pendulum and the mathematical
relationship
of this factor. This type of pendulum will consist of a mass hanging
on a length
of string.

Factors which affect the period (T) of a pendulum:
- Length (L) of pendulum
- Angle of amplitude
- Gravitational field strength (g)
- Mass of bob

I predict that the period will be affected by the length of the
pendulum. An
increase in length will produce an increase in time. I based by
prediction on
the scientific theory I found in a physics text book:

The pendulum is able to work when the bob is raised to an angle larger
than
the point at which it is vertically suspended at rest. By raising the
bob, the
pendulum gains Gravitation Potential Energy or GPE, as in being
raised, it is
held above this point of natural suspension and so therefore is acting
against
the natural gravitational force. Once the bob is released, this
gravitational
force is able to act on it, thus moving it downwards towards its
original
hanging point. We can say therefore, that as it is released, the GPE
is
converted into Kinetic Energy (KE) needed for the pendulum to swing.
Once
the bob returns to its original point of suspension, the GPE has been
totally
converted into KE, causing the bob to continue moving past its pivot
point and
up to a height equidistant from its pivot as its starting point.

The same factors affect the pendulum on its reverse swing. GPE gained
after
reaching its highest point in its swing, is converted into KE needed
for it to
return back to its natural point of vertical suspension. Due to this
continuous
motion, the bob creates an arc shaped swing. The movement of the
pendulum
is repeated until an external force acts on it, causing it to cease in
movement.
The pendulum never looses any energy, it is simply converted from one
form
to another and back again.


I am therefore going to experimentally determine the relationship
between the
length of the pendulum and the period.
In the scientific theory, I found a formula relating the length of the
pendulum
to the period. It stated that:

P = 2 L
g

P = The period
g = Gravitational Field Strength
L = Length of string

This formula shows that L is the only variable that when altered will
affect the
value of P, as all the other values are constants.

The formula: P = 2 L
g

can be rearranged to produce the formula: P = 4 L
g

and therefore: P = 4
L g

As 4 and g are both constants, this means that P must be directly
proportional to L.

I can now say that the length of the pendulum does have an affect on
the
period, and as the length of the pendulum increases, the length of the
period
will also increase.
I will draw a graph of P against L. As they are directly proportional
to each
other, the predicted graph should show a straight line through the
origin:

Method
- I will firstly set up a clamp stand with a piece of string 50cm long
attached to it.
- A mass of 50g will be attached securely to the end of the string
- The mass will be held to one side at an angle of 45 degrees
(measured
with a protractor), and then released.
- A stop clock will be used to time taken for one full oscillation
- This will be repeated a number of times, each time shortening the
length of string by 10cm
- The length of the pendulum will be plotted against the period on a
graph.

NB. The final length of string and mass will be decided after my
preliminary
investigation.

Apparatus:
- Meter ruler
- Protractor
- Clamp stand
- G-clamp
- Stop clock
- String
- Mass

Diagram:

The following factors will be considered when providing a fair test:
- The mass will be a constant of 50g throughout the experiment
- Angle of amplitude shall be a constant of 45 degrees. This will
ensure
that there is no variation of the forces acting on the pendulum.
- The value of gravitational field strength will inevitably remain
constant,
helping me to provide a fair test.
- The intervals between the string lengths will increase by 10cm each
time. This will help me to identify a clear pattern in my results.
- If any anomalous results are identified, readings will be repeated.
This
will ensure that all readings are sufficiently accurate.
- To ensure that the velocity is not affected, I will ensure that
there are no
obstructions to the swing of the pendulum.

The following factors will be considered when providing a safe test:
- Care will be taken not to let the bob come into contact with
anything
whilst swinging the pendulum, as the weight is relatively heavy (50g)
- The clamp stand will be firmly secured to the bench with a G-clamp
so
that the clamp stand will not move, affecting the results.
- Excessively large swings will be avoided (angle of amplitude will be
45
degrees

Results of preliminary investigation:

Length of string (cm) Period (secs)

50 2.58

40 2.31

30 2.11

20 1.78

10 1.39


My preliminary investigation was successful. The results from my table
back
up my prediction that, as the length of the pendulum increases, the
period
increases.

I learned from my preliminary investigation that my proposed method
may
not give me sufficiently accurate results. These results may be
inaccurate due
to a slight error of measurement in time, height or length. Although
this
experiment produced no anomalies, I will take three readings of each
value
during my final experiment and take an average. I will also measure
the time
taken for 5 oscillations rather that 1 and then divide the result by
5. These
two changes will hopefully help me to identify and eliminate
anomalies,
should they occur. They should also add to the accuracy of my results.

Obtaining Evidence

I used the method proposed in my plan, taking three readings of each
value
and measuring the time taken for 5 oscillations rather than for 1.
During the
experiment, I observed that each oscillation for the same length of
string
seemed to be equal. This showed that the pendulum did not slow down as
the
number of oscillations increased. I took the safety measures described
in my
original plan.

During the experiment I was careful to use accurate measurements in
order to
obtain sufficiently accurate results, for example:
- The string was measured with a meter ruler, to the nearest mm, to
ensure that each measurement had a difference of exactly 10cm.
- The angle of amplitude will be measured with a protractor to the
nearest degree to ensure that the angle remains constant throughout
the experiment.
- A stop clock will be used to measure the period accurately. The
period
was measured in seconds, with the stop clock measuring to the degree
of two decimal places of a second. However, I have rounded up each
time to the nearest second to give appropriate results.
- The mass was measured using five10g masses, to ensure that the mass
remained constant throughout the experiment.

Results:
Length of string (cm) Period (secs)
50 7.2
8.1
6.45

40 6.25
6.6
6.4

30 5.6
5.2
6.15

20 4.55
4.5
4.6

10 2.95
3.25
3.0

I took three readings of each value and took an average for each
concentration.
I then divided by 5 to get the average reading for one oscillation.
This again
should influence the accuracy of my results.

Table of averages:
Length of string (cm) Period (secs)
50 1.45

40 1.28

30 1.13

20 0.91

10 0.61

Using the formula, T = 2 L
g
found in the Scientific Theory, I calculated the perfect results that
should have
been obtained, had my experiment followed the formula exactly:
Length of string (cm) Period (secs)_
50 1.44

40 1.25

30 1.07

20 0.91

10 0.64
Using my averaged results, I squared P to show the relationship
between P
and L:

Length of string (cm) Period (secs)_
50 2.1

40 1.64

30 1.28

20 0.83

10 0.37

As all my results were accurate, I had no need to repeat any of them.
However,
had there been an anomalous result, or had I come across any problems,
I
would have repeated my results to identify the cause and eliminate
anomalies.


Analysing evidence and concluding

Using the results from my table, I drew a graph to show what had been
obtained from the experiment (see graph A). The graph clearly shows a
smooth curve with a positive gradient. This indicates that as the
length of the
pendulum is increased, the period will increase.

Although my second graph (see graph B), does not show a perfect
straight line
through the origin, a line of best fit can be drawn to show this. This
backs up
the theory in my scientific knowledge, that P is directly proportional
to L,
i.e. if the length of string was doubled, the period would be doubled.

My table of results drawn from my experiment was extremely similar to
the
results produced from the scientific formula, showing that my
experiment was
successful. My two graphs showed resemblance to my predicted graphs,
indicating that my results were sufficiently accurate and therefore,
my
proposed method was reliable for this experiment.

My findings indicate that the time period varies directly with the
length of the
string when all other factors remain constant.


Evaluating

The evidence obtained from my experiment supported my prediction that
as
the length of the pendulum increases, the period increases. This is
also shown
in Graph A, as the graph displays a smooth curve with a positive
gradient. My
method in squaring P was successful, as I discovered that T was
directly
proportional to L, providing all other values remain constant. This
was shown
by a straight line going through the origin (Graph B). These results
were
encouraging and led me to believe that my proposed method was
sufficient for
the experiment.

Some of the results were not accurate, as they did not match the
results
produced by the formula. This could have been due to human error.
However,
the majority of my results were no more than a decimal place away from
the
formula results and, therefore, quite reliable. Had there been any
anomalous
results, I would have repeated my readings.

Factors which may have affected the accuracy of my results include:
- Error in measurement of angle of altitude. This angle proved
difficult to
measure and it was hard to get the exact same angle for each result.
To
improve the accuracy of this measurement, I could have attached the
protractor to the clamp stand so that it was in a fixed position.
- Error in measurement of string. To improve the accuracy of this, I
could have marked off the points with a pen to ensure they were as
accurately measured as possible.
- Human reaction time. Depending on human reaction time, the
measurement period time could have been measured inaccurately, due
to slow reactions when setting the stop-clock etc. This could have
been
improved by involving another person to aid me with my experiment,
for a quicker reaction time.

The procedure was relatively reliable, excluding human error, and so I
can
conclude that my evidence is sufficient to support a firm conclusion
that:

The only factor which affects the period of a simple pendulum is its
length. As the length increases, so does the period.

If I were to extend my investigation, I would investigate to provide
additional
evidence to back up my conclusion, for example, changing the mass or
angle
of altitude. The results gained would hopefully aid me further in
supporting
my Scientific Theory. It would also be interesting to investigate how
the
factors are affected when the Gravitational Field Strength is
different, i.e.. not
9.8 Newtons.



the factors which affect the time for one full oscillation of a
pendulum
===============================================================

An investigation on the factors affecting the period of one complete
oscillation of a simple pendulum

In this investigation I aim to discover and investigate the factors
which affect the time for one complete oscillation of a simple
pendulum. It is important to understand what a pendulum is. A simple
pendulum is a weight or mass suspended from a fixed point and allowed
to swing freely. An oscillation is one cycle of the pendulums motion
e.g. From position a to b and back to a. The period of oscillation is
the time required for the pendulum to complete one cycle of its
motion. This is determined by measuring the time required for the
pendulum to reoccupy a given position.


I am going to do a simple preliminary experiment to investigate which
of the factors I test have an effect on the time for one complete
oscillation. The factors basic variable factors I can test are:
? Length (the distance between the point of suspension and the mass)
? Mass (the weight in g of the item suspended from the fixed point)
? Angle (the angle between the point of equilibrium and the maximum
point the pendulum reaches)
*The point of equilibrium is the point at which kinetic energy (KE) is
the only force making the mass move and not gravitational potential
energy (GPE).
I will test the extremes of these factors as I can assume that if they
have any effect on the period of oscillation it will become obvious.
To make sure my results are reliable and to allow for any anomalies I
will repeat the experiment 4 times for each extreme. I will also keep
all the other factors constant so if the results change for the
different extremes I can be sure which factor is causing this change,
as all the others will remain constant. To keep the results as
accurate as possible I will measure the period of 10 oscillations and
only use one decimal place to allow for my reaction time.

Results

Angle (º) Time Taken (sec) for 10 oscillations
90º 13·4 13·4 13·8 13·7 Average:13·6
45º 13·2 13·2 13·1 12·9 13·1
Length: 0·3m, Mass: 20g

Mass (g) Time Taken (sec) for 10 oscillations
400g 11·1 11·3 11·3 11·4 Average:11·3
100g 11·6 11·1 11·0 11·2 11·2
Length: 0·2cm, Angle: 45º


Length (cm) Time Taken (sec) for 10 oscillations
0·25m 10·4 10·5 10·5 10·3 Average:10·4
0·65m 16·8 16·0 16·5 15·9 Average:16·3
Mass: 50g, Angle: 40º
I can see from the results that there is one clear factor, length. For
Angle and mass the period for 10 oscillations is roughly the same for
both of the extremes. The variation between the averages is small
enough for me to conclude that these factors have a minimal effect if
any on the period of an oscillation. From the information from this
preliminary experiment I can now go onto investigate how precisely
length effects the oscillation period of a pendulum. I have also
learnt from this preliminary it is necessary for the clamp stand to be
held firmly in place so it does not rock.

Scientific Theory
As a pendulum is released it falls using GPE which can be calculated
using mass (kg) x gravitational field strength (which on earth is 10
N/Kg) x height (m). As soon as the pendulum moves this becomes KE
which can be calculated using 1/2 x mass (kg) x velocity2 (m/s2) and
GPE. At the point of equilibrium the pendulum just uses KE and then it
returns to KE and GPE and finally when the pendulum reaches maximum
rise it is just GPE and this continues. From this I can deduce that KE
= GPE. If these were the only forces acting on the pendulum it would
go on swinging forever but the energy is gradually converted to heat
energy by friction with the air (drag) and with the point the mass is
hung from. The amplitude of the oscillation therefore decreases until
eventually the pendulum comes to a rest at the point of equilibrium.
From this I can now explain why the amplitude and the mass have no
effect on the period of oscillation.

As the amplitude is increased so too is the GPE because the height is
increased which affects the GPE and therefore the KE must also
increase by the same amount. The pendulum then oscillates faster
because height or distance is involved in v2 in the KE formula.
However the pendulum has a larger distance to cover so they balance
each other out and the period remains the same. The period is also the
same if the amplitude is reduced.
For the mass as it is increased this affects both the GPE and the KE
as they both contain mass in their formulas but velocity is not
affected. The formulas below show that mass can be cancelled out so it
does not affect the velocity at all.
GPE = KE
mgh = 1/2mv2

Length affects the period of a pendulum and I have found a formula to
prove this and I will now attempt to explain it. The formula is:
T=period of one oscillation (seconds)
p=pi or p
l=length of pendulum (cm)
g=gravitational field strength (10m/s on earth)

This shows that the gravitational field strength and length both have
an effect on the period. However although the 'g' on earth varies
slightly depending on where you are as the experiments are all being
done in the same place this will have no effect as a variable. Length
is now the only variable. This means that T2 is directly proportional
to length.
T2= 2pl
g


The distance between a and b is greater in the first pendulum. However
the pendulum has gained no amplitude so therefore no additional GPE or
KE so it will still travel at the same speed. The first pendulum
therefore has a greater distance to travel and at the same speed so it
will have a greater period.
I can predict from this scientific knowledge that the period squared
will be directly proportional to the length.

Apparatus:
? Stopwatch
? Weights
? Ruler
? Protractor
? G-clamp
? Clamp stand
? String

Method: The apparatus was set up as shown above. The amplitude was
always 45ºand the mass 10g. I held the string taut and started the
stopwatch when I released the pendulum; I then stopped the stopwatch
after the tenth oscillation. I used a range of 10cm to 100cm to use a
suitable range of measurements. I also repeated each length 4 times to
make the average gained more reliable and to allow for any anomalies.
To make the results more accurate I also counted 10 oscillations
meaning if you divide the period by ten your reaction time, which
affects the length of the period, is reduced by 9/10. To make sure
this was as fair a test as possible I:
? Tried to create as little friction as possible where the string is
attached to the clamp.
? Let go with out adding any extra forces
? Kept the string taut
? Made sure the mass and angle remain the same in case they have a
small effect on the period.
? Keep the whole experiment in the same place so that the
gravitational field strength does not change
To make this a safe experiment:
? No weight above 400g
? No angles above 90º
? The clamp stand is secure

Results:
A Table to show the periodic time for 10 oscillations for various
lengths

Periodic time (seconds)
Length (m)
0·1 6·1 6·1 6·1 6·0 Average periodic time: 6·08
0·2 8·6 8·7 8·6 8·7 Average periodic time: 8·65
0·3 10·2 10·3 10·3 10·3 Average periodic time: 10·28
0·4 12·6 12·4 12·6 12·3 Average periodic time: 12·48
0·5 14·0 14·1 13·9 13·8 Average periodic time: 13·95
0·6 15·2 15·3 15·3 15·3 Average periodic time: 15·28
0·7 16·4 16·4 16·5 16·6 Average periodic time: 16·48
0·8 17·8 17·6 17·5 17·7 Average periodic time: 17·65
0·9 18·8 18·5 18·4 18·8 Average periodic time: 18·63
1·0 19·6 19·3 19·3 19·3 Average periodic time: 19·38
Angle: 45°, Mass:10g
To find the average period of one oscillation I must divide my average
for 10 oscillations by 10.
A table to show the period of one oscillation for various lengths

Length Period Period according to formula
0·1 0·61 0·63
0·2 0·87 0·89
0·3 1·03 1·09
0·4 1·25 1·26
0·5 1·40 1·40
0·6 1·53 1·54
0·7 1·65 1·66
0·8 1·77 1·78
0·9 1·86 1·88
1·0 1·93 1·99

I am now going to create a table by rearranging the formula I found so
length is directly proportional to the period of oscillation squared.

A table to show the period squared for various lengths

Length Period squared Period squared according to formula
0·1 0·37 0·40
0·2 0·76 0·79
0·3 1·06 1·19
0·4 1·56 1·59
0·5 1·96 1·96
0·6 2·34 2·37
0·7 2·72 2·76
0·8 3·13 3·17
0·9 3·46 3·53
1·0 3·72 3·96

The above two tables are to two decimal places so that the data is
easier to draw a graph from. I did not include the period or the
period squared on my graphs because the results are too close together
but the tables indicate how near my results were to what they should
be in theory.
From my results I have found out that the period squared is as
predicted directly proportional to the length of the pendulum because
my graph is a straight line and goes through 0. Also if you take the
length at 0·1m the period squared is 0·37 and then if you take the
length at 0·5m the period squared is 1·96m. The point of this is to
show that that both the period and the length go up by nearly exactly
the same proportion because 0·5/0·1=5 and 1·96/0·37=5·3. The graph
with period plotted against length also provides the useful
information that period and length have a relationship, which involves
the indice 2 . I have noticed the pattern that if you divide the
period squared of the pendulum by the length of the pendulum you get
roughly the same figure each time and that the ratio between length
and the period squared is roughly 1:38.
I can draw a conclusion from my evidence that the formula:
This is correct because if you rearrange it to form T2= 2pl this fits
perfectly
g
with the graph and my results. If you remove the constants from the
formula you are left with a direct link from T2 to l. The curve also
reinforces the original formula by showing that as l increases by 0·1
the period increases by a much larger percentage. As the length
increases the period goes up in smaller and smaller amounts, which
again agrees with the formula. These results totally support my
original prediction and they also support the scientific theory. The
shape of the graph immediately shows this.

The results obtained show that my experiment was successful for
investigating how length effects the period of an oscillation because
they are the same and agree with what I predicted would happen. The
procedure used was not too bad because my results are very similar to
what they would be under perfect circumstances. My results are
reasonably accurate as they fulfil what I thought and said would
happen. However there are a few minor anomalies which can be seen in
the graphs and in the tables. They have a larger gap from what they
should have been according to the formula than usual. At 1m there is
an anomaly which is a few fractions of a second away from the line of
best fit.
Most of the procedure was suitable because it gave a useful and
relevant outcome but it could have been improved in a number of ways.
The reliability of the evidence could be increased by making the angle
more precise, making sure the string is taut when the pendulum is
released and making the string the exact length it should to be. The
anomalous results I have may be down to a number of reasons but could
mainly be blamed on my releasing the pendulum and providing it with an
external force, which would affect the period. My timing of the
stopping and starting of the stopwatch could be inaccurate. The
overall results may be a 1/100 of a second out because I used the
gravitational field strength of 10 when the actual field strength may
be different. If the above improvements were added in, the results
would be more accurate and reliable. To further this work, I would
repeat each length providing more accurate averages. I would provide
additional evidence for my conclusion by increasing the range of
lengths and decreasing the intervals between the lengths to five
centimetres. These additions would extend my investigation further.

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