# 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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More ↓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 pâˆštex2html_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 âˆštex2html_wrap_inline105/g

T = 2p âˆš0.3/9.8

T = 1.009s

Length 0.4, g-force = 9.8N/KG

T= 2pâˆštex2html_wrap_inline105/g

T = 2p âˆš0.4/9.8

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 90Â°). 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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2. Angle of release - I predict that the more the angle the less the

period of time, but this only applies up to 90Â°

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 10-90Â°,

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 Î± 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 Î¸

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 = Â½mvÂ²

The mass cancels out from both sides of the equation leaving, gh = Â½vÂ²

So:

vÂ² = 2gh

v = âˆš2gh

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

Ã˜ 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.

Ã˜ 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.

Ã˜ When doing the angle of amplitude experiment I found few anomalous

results which could have been due to the increments of 1 degree.

Ã˜ 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.

Ã˜ 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= 2Ï€âˆštex2html_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= 2Ï€âˆštex2html_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.