Investigating the Factors that Affect the Period of One Swing of a Pendulum

Investigating the Factors that Affect the Period of One Swing of a Pendulum

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Investigating the Factors that Affect the Period of One Swing of a Pendulum

Aim: To investigate the factors which affect the period of one swing
(oscillation) of a simple pendulum. The factors I will use are length
of the string, and angle that the bob is released from.

Hypothesis:

1. Length of string

I think that the length of the string directly affects the period of
one oscillation. The mathematical formula used to describe the period
of the pendulum is:

T= 2 p√tex2html_wrap_inline105/g

T is the period (time for one swing - seconds)

tex2html_wrap_inline105 is the length of the pendulum (metres)

g is the acceleration dues to gravity. (N/KG)

tex2html_wrap_inline105 (Length) is in the formula, clearly indicating
that it is a factor which will directly affect the period of time.

To see whether the time period will increase or decrease when the
length is increased, I will substitute the formula for numbers to see
the result.

Length 0.3, g-force = 9.8N/KG

T= 2p √tex2html_wrap_inline105/g

T = 2p √0.3/9.8

T = 1.009s

Length 0.4, g-force = 9.8N/KG

T= 2p√tex2html_wrap_inline105/g

T = 2p √0.4/9.8

T = 1.269s

The calculations above show that when the length of the pendulum is
0.3m, the time for one oscillation is 1.009s. When the length is
increased, the time is increased. When length is 0.4m, time period is
1.269s.

This tells us that when the length is increased, the time period is
increased.

2. Angle of release

A simple pendulum is only a weight known as a "bob" hung from a
string. When the bob is lifted, the pendulum gains potential
gravitational energy, as it is acting against the force. Therefore,
the angle, which would raise the height, would give the bob more
gravitational energy (up to 90°). The more the angle, the more the
energy, the faster the swing, the less the period of time.

Prediction:

1. Length of string - I predict that the longer the length of string,

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the longer the period of one oscillation.

2. Angle of release - I predict that the more the angle the less the
period of time, but this only applies up to 90°

Apparatus:

* Clamp stand

* String

* Bob

* String

* Protractor

* Ruler

* Stopwatch

Method

* Set up the clamp stand, quite high on a stand and place at the
edge of the table. Tie a string with a bob at the end (a simple
pendulum) onto the clamp.

* Allow the pendulum to swing freely, and ensure that the stand is
steady on the table. If it is shaking, the weight may allow the
pendulum to move faster, or slow it down, and this could affect
our results.

* Attach a protractor to the clamp so that the angle of release can
easily be measured. This is needed for all experiments because the
angle has to be constant in the other experiments.

* Swing the pendulum from a particular angle of release, and make
sure that the same angle is used for the whole experiment.

* Start the stopwatch when the bob is released, in order to measure
the time period of 10 oscillations. 10 is a good number to choose,
because it is not too small, or the results may not be accurate.

* Repeat the experiment again, changing the independent variable
every time. In the case of length, I did the ranges
5-50, going up in 5s, and in the angle of release I did 10-90°,
going up in 10s.

Variables

1. Length of string

Independent: Length of string

Dependant: time period of 10 oscillations

Control:

* Angle of release

* Height from ground

* Position of pendulum

* Mass of bob

2. Angle of release

Independent: Angle of release

Dependant: Time period of 10 oscillations

Control:

* Length of string

* Height from ground

* Position of pendulum

* Mass of bob

Conclusion

A pendulum is something hanging from a fixed point which, when pulled
back and released, is free to swing down by gravity and then out and
up because of its inertia, or tendency to stay in motion. A simple
pendulum consists of a mass (called the bob) attached to the end of a
thin cord, which is attached to a fixed point. When the mass is drawn
upwards and let go, the force of gravity accelerates it back to the
original position. The momentum built up by the acceleration of
gravity causes the mass to then swing in the opposite direction to a
height equal to the original position. This force is known as inertia.
A period is one swing of the pendulum over and back. The frequency is
the number of back and forth swings in a certain length of time.

Experiment 1

By looking at the results obtained from by experiment I found that as
the length increases the period of 1 oscillation increases too (this
can be see by the line of the graph going up). The graph's gradient is
T2=4Ï€tex2html_wrap_inline105/g. the results obtained from my graph
matches the result I calculated from my theoretical prediction. When
looking at my graph I found no anomalous results.

If I was to compare the theoretical prediction graph to my graph of
the actual result it does not show a perfect straight line through the
origin, thus, a line of best fit can be drawn to show this. This will
therefore justify my prediction and that T is directly proportional to
tex2html_wrap_inline105, (so, if the length of string was to be
doubled, the period would be doubled as well).

The statement tex2html_wrap_inline105 α T can be justified by taking
values from the graph, for example when the length of the string is
5cm T= 0.481 and when the length is doubled to 10cm T= 0.651, which
shows T is almost doubled.

The table below shows actual results compared to the theoretical by
working out the percentage error by this formula, percentage error=
(actual error (actual result- theoretical results)/ exact value
(theoretical results)) x 100:

Length of string (cm)

Theoretical prediction

Actual

results

Percentage error

5

0.449

0.481

7%

10

0.635

0.651

2.5%

15

0.777

0.862

11%

20

0.898

0.877

2%

25

1.004

1.014

1%

30

1.099

1.129

2%

35

1.187

1.221

5%

40

1.269

1.267

0.2%

45

1.345

1.328

1%

50

1.419

1.489

5%

My average percentage error is 3.6% which suggests that our results
are fairly accurate.

Experiment 2

By looking at the results obtained from my graph I found that the
angle of amplitude did affect the period of oscillation, however in a
very slow rate. Also I found some anomalous results in this experiment
which could have been because we did not follow one of our control
variables.

Experiment 3

By looking at the table and graph obtained from my results I found
that by increasing the mass of the bob had no effect to the period of
one oscillation. This could be because that since the gravitational
acceleration is 9.8N at all time (on Earth); the mass of the bob will
have no deciding effect on the period of oscillation. The reason for
this can be taken from my prediction, which is:

[IMAGE]


Height =tex2html_wrap_inline105 - tex2html_wrap_inline105 cos θ

To explain the fact that mass does not affect the period of one
oscillation in a simple pendulum, we need to use these equations:

Potential Energy (PE) at (any point) = m ( mass of the bob) x g x h

Kinetic Energy at position 1 = Potential Energy at position 2

Potential Energy at position 1 = Kinetic Energy at position 2

Hence, PE = mgh

Potential Energy at position 2 = Kinetic Energy at position 1

Thus, mgh = ½mv²

The mass cancels out from both sides of the equation leaving, gh = ½v²

So:

v² = 2gh

v = √2gh

From looking at the equation above we can soundly say that the speed
of the bob is not in any way affected by the mass.

Evaluation

Ø My experiment on how the length of the pendulum affected the period
of one oscillation was successful, since by looking at my graphs I
detect no anomalous results. So my method of squaring P was thus
correct.

Ø Some of the results that were obtained from the length experiment
were not accurate, since it did not match the results produced by my
theoretical prediction completely. The reasons for the outlined could
because of:

o Human error. However, the majority of my results were no more than
few decimal places away from the formula results and, thus, quite
reliable.

o Error in measurement of angle of amplitude (increments of 1 degree):
to improve this we could have ensured that there were two protractors
(one in front and the other behind the pendulum).

o Error in measurement of string (marginal error of 0.5mm from both
sides): this could have been improved by marking the exact measurement
on the string and ensure that the string is hanging from there
exactly.

Ø When doing the angle of amplitude experiment I found few anomalous
results which could have been due to the increments of 1 degree.

Ø When doing the changing the mass experiment I found that it had no
effect on the period of one oscillation of a pendulum which was
successful. The mass thus (as shown in my graph) played no part in
changing the period of a pendulum and the graph showed a constant
line. Now we can eliminate mass as a factor that affects the period of
oscillation in any way.

Ø To promote further investigation it will be interesting to observe
how the period of a pendulum oscillation is affected when the
Gravitational Field Strength is different (i.e. not 9.8 Newtons).

o The aim of this experiment will be to investigate theoretically how
a change in the gravitational strength in the equation T= 2π√tex2html_wrap_inline105/g
affects the period of one oscillation in a simple pendulum.

o The hypothesis in this experiment will be, "as the gravitational
strength is made higher the period of one oscillation is lower"

o Prediction

I can now investigate theoretically how the gravitational field
affects the period of a simple pendulum; I can do this by keeping a
constant angle of amplitude and length of string. Below are lists of
gravitational field strength of different planets:

Planet

Gravitational Field Strength (N/Kg)

Mercury

4

Venus

9

Earth

9.8

Mars

4

Jupiter

26

Saturn

11

Uranus

11

Neptune

12

Pluto

4

Deep space

0

o By using the above information I can draw up a list of theoretical
predictions by substituting the gravitational field strength within
this equation T= 2π√tex2html_wrap_inline105/g. Hence I will be able to
work out the period (T) for the gravitational strength of each planet.
To work out the theoretical prediction I must keep the length as a
constant of 0.5m.

o Theoretical Prediction

Planet

Gravitational Field Strength (N/Kg)

Time period (T) (s)

Mercury

4

2.22

Venus

9

1.48

Earth

9.8

1.42

Mars

4

2.22

Jupiter

26

0.87

Saturn

11

1.34

Uranus

11

1.34

Neptune

12

1.28

Pluto

4

2.22

Deep space

0

Infinity

[IMAGE]

o By looking at the table I can now base my results on a sound
prediction and say that the stronger the gravitational field strength
is of a planet the faster the time period is of one oscillation and
the weaker the gravitational field strength the slower the time period
of one oscillation.

o I cannot continue this investigation, since my school does not have
the resources for me to experiment on other planets.

This controlled-falling system is a weight (bob) suspended by a string
from a fixed point so that it can swing freely under the influence of
gravity. If the bob is pushed or pulled sideways, it can't move just
horizontally, but has to move on the circle whose radius is the length
of the supporting string. It has to move upward from where it started
as well as sideways. If the bob is now let go, it falls because
gravity is pulling it back down. It can't fall straight down, but has
to follow the circular path defined by its support. This is
"controlled falling": the path is always the same, it can be
reproduced time after time, and variations in the set-up can be used
to test their effect on the falling behaviour.
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