Answer Add Mt 2008

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~Contents~

Introduction -------------------------------------------1

Aim ------------------------------------------- 2

Task Specification -------------------------------------------

3

Problem Solving -------------------------------------------4 15

Further Exploration -------------------------------------------16 19

Conclusion ------------------------------------------- 20

Acknowledgements -------------------------------------------21

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~Introduction~

Time is a component of a measuring system used tosequence events, to compare the durations of events and theintervals between them, and to quantify the motions of objects. Time has been a major subject of religion, philosophy, and

science, but defining time in a non-controversial mannerapplicable to all fields of study has consistently eluded thegreatest scholars.

There are many ways to measuring time; some ancientmethods include using a sundial, or an hourglass. A pendulum canalso be used as a basis for measuring time. A simple pendulumconsists of an object, usually a pendulum bob, suspended by astring from a fixed point. When displaces, and then released, thebob will swing back and forth in a vertical plane under the

influence of gravity. This periodic motion can be used as a basis todetermine time.In this project work, I will focus on using the simple

pendulum to measure time, with the help of some mathematicalknowledge.

Photo 1: A sundial is an ancient method

for measuring time

Photo 2: Another

way to measuretime is using anhourglass
http://en.wikipedia.org/wiki/Measurementhttp://en.wikipedia.org/wiki/Religionhttp://en.wikipedia.org/wiki/Philosophyhttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Measurementhttp://en.wikipedia.org/wiki/Religionhttp://en.wikipedia.org/wiki/Philosophyhttp://en.wikipedia.org/wiki/Science

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~Aim~

The aims of carrying out this project work are to:

i) Develop mathematical knowledge in a way whichincreases students interest and confidence;

ii) Apply mathematics to everyday situations and to begin tounderstand the part that mathematics play in the worldwe live;

iii) Improve thinking skills and promote effectivemathematical communication;

iv) Assist students to develop positive attitude andpersonalities, intrinsic mathematical values such asaccuracy, confidence, and systematic reasoning;

v) Stimulate learning and enhance effective learning

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~Task Specification~Identifying and stating all required information

The simple pendulum shown in Figure 1 is set in motion byreleasing the object through a small angle of displacement, (10to 15 ) from the vertical.

Procedure:

1. A simple pendulum is set up as shown in Figure 1 byattaching an object to a string of length 60cm.

2. The pendulum is set in motion and the time taken, t s ismeasured for 20 complete oscillations.

3. The period, T s, that is the time taken for one completeoscillation is calculated.

4. Steps (1) to (3) are repeated using at least 10 differentlengths of strings with the minimum length of 5 cm.

5. The readings are recorded in a suitable table.

Figure 1

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6. A graph of period (T s) is plotted against length ( ). Thegraph obtained is commented.

~Problem Solving~Results from the conducted experiment:

Length ofpendulum

, l (cm)

Number ofcompleteoscillation

(x)

Time taken for number ofcomplete oscillation, t(s)

1st

Attempt2nd

AttemptAverage

Period, T(s)

60 20 32 32 32.0 1.60055 20 30 31 30.5 1.52550 20 29 29 29.0 1.45045 20 27 27 27.0 1.35040 20 26 26 26.0 1.30030 20 23 23 23.0 1.15020 20 19 19 19.0 0.95015 30 25 25 25.0 0.83310 30 21 21 21.0 0.7005 30 17 17 17.0 0.567

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The Relationship between Period, T and length, l

The relationship between period and length is shown by the given

formula:

The gravitational acceleration, g is not a variable, it is a constant.Therefore the variables in the following formula are T and .

(a)Suggest at least two pairs of variables for the horizontal andvertical axes to obtain a linear relation.For each pair, plot the graphs to draw the lines of best fitmanually and by using ICT.

To obtain a linear equation, the equation above can besquared to form a linear equation.

The two pairs of variables are:(i) T and ( T as the y-axis and as the x-axis)(ii) and ( as the y-axis and as the x-axis)

In order to draw the graphs of these two pairs of variables, weneed to find the vales of and .

Period, T(s)

Length ofpendulum, (cm)

(cm)

1.600 2.560 60 7.746

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1.525 2.326 55 7.4161.450 2.103 50 7.0711.350 1.823 45 6.7081.300 1.690 40 6.325

1.150 1.323 30 5.4770.950 0.903 20 4.4720.833 0.694 15 3.8730.700 0.490 10 3.1620.567 0.322 5 2.236

Graph of against

Graph of against

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Drawing the graphs manuallyi. Graph of T against

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ii. Graph of against

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(b)Estimating the gradients of both graphs

From the graph T against , we can find that the equation is0.188x+0.115.From the formula , we can determine thegradient if we compare the formula to the equation obtained.

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Since 1116.663 is in cm s-2, we have to change it into m s-2 bydividing the value by (10 x 10 = 100). We get 11.17 m s -2. Thisvalue is bigger than the original value of 9.807 m s -2.

Value of obtained from the experiment = 11.17Value of actual value of = 9.807

Difference = 11.17 9.807= 1.363

Percentage of error =

= 13.898%

Comment:The value obtained from the experiment is higher by 1.363 fromthe actual value of g, which is 9.807. The reason is theexperiment not conducted in an enclosed system, therefore,many factors can influence the accuracy of the results obtained,such as air movement, and air resistance.

(ii) From the graph of against

or

= y-axis= x-axis

So gradient =

To find the value of gravitational acceleration, m s-2,

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Since 986.96 is in cm s-2, we have to change it into m s-2 by

dividing the value by (10 x 10 = 100). We get 9.87 m s -2. Thisvalue is a little bigger than the original value of 9.807 m s -2.

Value of obtained from the experiment = 9.87Value of actual value of = 9.807

Difference = 9.87 9.807= 0.063

Percentage of error == 0.642%

Comment:If compared with the first graph, the percentage of error is muchsmaller. The first value of obtained is slightly higher than theactual value of , but the second value of obtained is a bithigher than the actual value. This is probably caused by thepresence of air resistance. Another probable cause of the

deviation of the value from the actual value is that the pendulumbob did not oscillate in a plane but in a circle.

(d)Using the graph with the least percentage of error todetermine the length of the string that will produce acomplete oscillation in 1 second.

The graph with the least percentage of error is the graph showingagainst .

The relationship between period and length is:

When = 1,

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25.00cm is the length of string that will produce a complete

oscillation in 1 second.

Using a Simple Pendulum to Calculate the Pulse Rate

1. A simple pendulum is made out of a 25.00cm string ( toensure that 1 oscillation is equivalent to 1 second)

2. A friend is asked to count the number of oscillation.

3. He is asked to give instructions when to start and stop (after

30 oscillations).

4. The counting of the pulse is started when he says start andstopped when he says stop after 30 oscillations.

5. The process is repeated 3 times to get the value, Paverage.

6. The pulse rate = Paverage x 2.

Change in the Period, if the length of the string in increased by 4times.

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When l is increased by 4 times,

=

Substitute into the equation,

Since

Then,

Therefore, when the length of the pendulum is increased by 4times, the period increases by 2 times.

~Further Exploration~

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Determining the new period if the pendulum is set in motion onthe moon

Gravitational field strength of the moon, gmoon = 1.622ms-2

Period of pendulum on Earth, TEarth = 1 sGravitational acceleration of the Earth, gEarth = 9.807 ms-2

Period of pendulum on the moon, TMoon = TMoon

Technique of solving:Using simultaneous equation

Solve (1) and (2) simultaneously

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Investigating whether a pendulum will continuously swing in air

A pendulum cannot swing continuously swing in air due todamping. Damping is defined as loss of energy to the

surroundings, usually in the form of heat energy. In air, there is airresistance, which induces friction upon the swinging pendulumbob. This causes the energy in the pendulum bob to be loss in theform of heat energy, hence damping occurs. This results in thependulum swinging slower and slower. For the pendulum tobecome a perpetual machine (swing continuously), it has to swingin complete vacuum.

Conditions for the pendulum to swing continuously:- No hinge friction

- In a vacuum- No internal friction

Comparing the time taken for the pendulum to stop in water withthe time taken for the pendulum to stop in air.

Time taken for the pendulum to complete one oscillation in water:

T1 = 4s

T2 = 5sT3 = 4s

TAverage = 4.33s

Time taken for the pendulum to complete one oscillation in air:

T1 = 1.55sT2 = 1.32sT3 = 1.62s

TAverage = 1.50s

Based on the values above, the time taken for the pendulum tomake a complete oscillation in air is shorter, if compared to the

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time taken for the same pendulum to make a complete oscillationin water.This is due to the water resistance and the buoyant force of thewater acting on the pendulum bob, thus making the pendulum

bob swing slower, as the water resistance is stronger toovercome, compared to air resistance.Comparing the motion of the pendulum in air, water and in avacuum

In Air

In water

In Vacuum

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Comparing and contrasting the graphs

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~Conclusion~A pendulum can be used to measure time, as shown and proven

in the experiment I conducted. The period, time taken for acomplete oscillation can be calculated using the formula below:

The period of the pendulum swinging in air and in water varies, asthe water resistance is stronger, compared to air resistance.

The pendulum is not affected by gravity, but due to internalresistance, hinge resistance and air resistance, it is not able tooscillate continuously, i.e becoming a perpetual machine.

As shown in the experiment, the pendulum is quite a reliableinstrument to calculate time, but is not used, as the process ofcalculating time using a pendulum is very complex.

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~Acknowledgements~

I would like to express my gratitude and thanks to myteacher, ________ for her wonderful guidance for me to be able tocomplete this project work, to my parents for their continuoussupport to me throughout this experiment, to my friends for theirhelp, and to all those who contributed directly or indirectlytowards the completion of this project work.

Throughout this project, I acquired many valuable skills, and hopethat in the years to come, those skills will be put to good use.

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Answer Add Mt 2008