ADD MATH PROJECT (FORM 5) 2008

A simple pendulum consists of an object suspended by a string from a fixed point.When displaced, and then released,the object will swing back and forth in a vertical plane under the influence of gravity.This periodic motion can be used as a basis for measuring time.

The simple pendulum as shown in Diagram 1 is set in motion by releasing the object through a small angle of displacement, θ(10°to 15°)from the vertical.

Procedure1) Set up a simple pendulum by attaching an object to a string of length 60cm.2) Set the pendulum in motion and measure the time taken, t s,for 20 complete oscillations.3) Calculate the period,Ts,that is the time taken for one complete oscillation.4) Repeat steps 1 to 3 using at least 10 different lengths of strings with the minimum length of 5 cm5) Record the readings in a suitable table.6) Plot a graph of period (Ts) against length (l cm).Comment on the graph obtained.7) The relationship between period and length is given as

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,where g is the gravitational acceleration.

<!--[if !supportLists]-->a. <!--[endif]-->Suggest at least two different pairs of variables for the horizontal and vertical axes to obtain a linear relation.For each pair,plot the graphs and draw lines of best fit manually and by using ICT.

<!--[if !supportLists]-->b. <!--[endif]-->Estimate the gradient of each graph.Hence,write an equation relating period and length for each of the graphs.

<!--[if !supportLists]-->c. <!--[endif]-->Use the gradient of each graph to determine the respective value of the gravitational acceleration,g ms-2.Comment on the values obtained.How do these values of g compare with the accepted value of g on earth(9.807 g ms-

2).Calculate the percentage error for each of the value of g obtained.Explained the difference(if any).

<!--[if !supportLists]-->d. <!--[endif]-->Use the graph with the least percentage error in g to determine the length of string that will produce a complete oscillation in 1 second.

A simple pendulum can be used as a device to measure time.Describe how you can use it to measure your pulse rate.9) If the length of the string is 4 times its original length,state the change in the period,Ts.

Further Exploration1) If a simple pendulum with a period of 1 second is set in motion on the moon,determine the

new period of this pendulum.2)

<!--[if !supportLists]-->a. <!--[endif]-->Investigate whether a simple pendulum will swing continuously in air.Explain your findings.Suggest the conditions required for a pendulum to swing continuously.

<!--[if !supportLists]-->b. <!--[endif]-->If a pendulum is made to swing in water,compare the time taken for this pendulum to come to a complete stop with the time taken by a pendulum swinging in air.Explain the difference.

3) Sketch graphs on the same scale to illustrate the motion of a simple pendulum swinging

<!--[if !supportLists]--> i. <!--[endif]-->in air,

<!--[if !supportLists]--> ii. <!--[endif]-->in water and

<!--[if !supportLists]--> iii. <!--[endif]-->in vacuum.

Compare and contrast the graphs

Well, this is more like a physics question than an add maths question. <!--[if gte vml 1]&gt; &lt;![endif]--><!--[if !vml]--><!--[endif]-->

I think the problem most students facing are: they don’t have the data to start! Well, if you don’t want to trouble your lab assistant to prepare the apparatus for each of you and your friends to carry out the experiment, there is alternative for you: the java applet simulation.

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Here I have found a good one. The Java Applets on Physics created by Walter Fendt. You will need a Java plug-in to run the applet. If your browser can’t run the applet, you will need to download and install one.

If you have the Java Plug-in in your browser, you will see something like this. Follow the following steps to carry out the experiment.

<!--[endif]-->Walter Fendt Java Applet of Simple Pendulumpendulum.png (8.52 KB) Viewed 19403 times

1. The first thing that you need to do is to select your length. 2. By default, the length is 5.0m. This is obviously too large for this project because you

are unlikely to use this length (about 2 storey high) to do the experiment. Let’s start with 0.5m (50cm), this is the minimum length allowed in this applet. Leave the gravitational acceleration, the mass and the angle to its default value.

3. Click on the “start” button, count for 20 complete oscillation, and then click “pause” 4. Record the time (at the lower left corner). 5. Click on the “reset” button. Repeat the experiment with other value of length (Any

value within .05m to 1.50m will be acceptable).

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Take note that 1 complete oscillation is the oscillation to and fro back to its initial position as shown in the diagram below.

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Oscillation from point A to point B is considered to be 1 complete oscillationgraph.png (1.99 KB) Viewed 19347 times

The following table shows the result that I obtain from this applet. You are encouraged to do it by yourself.

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I’ve use the above data to plot the graph of Period against Length (Show in the figure [graph 1] above) and found that the result is not satisfying . The data shows that the graph looks like

a straight line whereas it should be a curve. Even though I’ve draw it as a curve but it doesn’t look nice. This is mainly due to the range of the length that I use is too small.

I’ve try to repeat the “experiment” with another set of values of the length. The first 2 reading (20cm and 40cm) are self-created because the minimum length allowed in the applet is 50cm. The result is as below.

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The graph looked much better (graph2 below). Therefore all the subsequent calculation are base on the second set of data.

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The relationship between the period and the length is

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The gravitational acceleration g is constant (not variable). Therefore the variable is T and √l.

If we square both sides of the equation, it become

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or

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Hence the 2 variables are T2 and l.

The 2 pairs of variables are:

1. T and √l (T as verticle axis and √l as horizontal axis) 2. T2 and l (T2as verticle axis and l horizontal axis.)

In order to draw the graphs with these 2 pairs of variables, we need to find the values of T2 and √l. The data is as below:

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Well, now we can start to plot our linear graph. I leave the “manual” part to you. Here I will only discuss “draw lines of best fit ….. by using ICT.”

There are many softwares you can use to plot a graph. I found a simple one in internet. Again, it’s a java applet for discussion on Regression and Correlation created by the National Council of Teachers of Mathematic (NCTM).

It’s easy to use. You just need to input the minimum and maximum scale for the x-axis and y-a-xis, and then press the “set scale” button to set your scale, and start to plot your points by clicking on the graph. After finish plotting your points, press the “show line” button to draw the line of best fit. The equation of the line is given in the window at the upper right corner.

My results are as follow:

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Graph of T against √lgraph3.png (6.63 KB) Viewed 19022 times

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Graph of T² against l.

Graph of T against √lFrom the applet, we found that the equation of the graph T against √l (The 1st graph) is y=0.201x + 0.00243, which means the gradient of the graph is 0.201. The equation relating the period and the length is

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Graph of T² against lThe equation of the graph T² against l is y=0.041x-0.0224. The gradient of the graph is 0.041. The equation relating the period and the length is

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c. Use the gradient of each graph to determine the respective value of the gravitational acceleration,g ms-2.Comment on the values obtained.

How do these values of g compare with the accepted value of g on earth(9.807 g ms-2).Calculate the percentage error for each of the value of g obtained.Explained the difference(if any).

Graph of T against √l

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The y-axis = TThe x-axis = √l, henceThe gradient = 2π/√g

From the graph, we know that the gradient is equal to 0.201. Therefore

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CommentThis value is slightly lower than the accepted value 9.807 ms-2.

Percentage of ErrorThe error is 9.772 - 9.807 = -0.035 ms-2

Percentage of error

Graph of T 2 against l

The y-axis = T2

The x-axis =l, henceThe gradient = 4π2/g

From the graph, we know that the gradient is equal to 0.201. Therefore

CommentThis value is slightly lower than the accepted value 9.807 ms-2.

Percentage of ErrorThe error is 9.629 - 9.807 = -0.178 ms-2

Percentage of error

Explanation the DifferencesBoth of the values that we got is slightly lower than the accepted value. This is most probably due to the presence of air resistance. This error can be reduced by reducing the angle of oscillation.

Another possible source of error is the pendulum did not oscillate in a plane but in circle. This make the pendulum become a cone pendulum, where the calculation will be different from a simple pendulum.

d. Use the graph with the least percentage error in g to determine the length of string that will produce a complete oscillation in 1 second.

The graph with the least percentage error is the graph of T against √l, the relationship between the period and the length is

.

When T = 1,

A simple pendulum can be used as a device to measure time.Describe how you can use it to measure your pulse rate.

1. Make a simple pendulum of length 24.75cm (So that 1 oscillation is equivalent to 1 second).

2. Get a friend to count the number of oscillations for you. 3. Ask him to give instruction when should you start and stop (after 30 oscillation). 4. Start counting your pulse when your friend says start and stop counting after 30

oscillation. 5. Repeat this process for 3 times to get the average value, Pave.

The pulse rate = the Pave x 2.

9) If the length of the string is 4 times its original length,state the change in the period,Ts

When the length is 4 times its original length,

l’ = 4l

Substitute l’ into the equation,

Since

Conclusion:If the length of the pendulum increases by 4 times, the period, T will increase by 2 times.

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