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Data TableThe data table below shows the raw data collected which includes the decreases in thelength of the string and the time trials for ten oscillations

Length of string (m)Time for 10 oscillations (s)Trial 1Trial 2Trial 3Average time(s)0.53515.1215.2415.1915.18 ± 0.060.50214.6914.5314.8114.68 ± 0.140.43513.8414.3214.0714.08 ± 0.390.36512.6212.4412.5312.53 ± 0.090.31711.8911.9011.7211.84 ± 0.090.23510.5910.4310.5610.53 ± 0.080.1659.199.129.159.15 ± 0.030.1258.418.097.978.16 ± 0.22± 0.001m±0.01s±0.1s

The table below shows the values of the calculated data that includes the time period(time for one oscillation) which further gives us the acceleration due to gravity.

Average Time(s)Time Period(s)Acceleration due toGravity (ms

-2

)15.18 1.529.14 ± 114.68 1.479.17 ± 114.08 1.418.64 ± 112.53 1.259.22 ± 111.84 1.188.99 ± 210.53 1.058.42 ± 29.15 0.927.70 ± 18.16 0.827.34 ± 2±0.1s±0.1s± 1ms

-2

Data ProcessingAverage Time (s) = Trial 1 + Trial 2 + Trial 33= 15.12 + 15.24 + 15.193= 15.18 s

Uncertainty for Average = (Highest value – Lowest value)2= (15.24 – 15.12)2= ±0.06sTime for one oscillation = Time for 10 oscillation10= 15.1810= 1.52 s

T =

g = 4π

2

l T

2

=

4π

2

(0.535)(1.52)

2

= 9.14 ms

-2

Uncertainty of gravity:

(Uncertainty 1) + (Uncertainty 2) X 100(Value 1) (Value 2)

0.2 + 0.001 X 1001.52 0.535= 9.14 ms

-2

±13.3 %= 13.3 X 9.14100= ±1.21562= 9.14 ms

-2

± 1 ms

-2

Average of Gravity = 9.14 + 9.17 + 8.64 + 9.22 + 8.99 + 8.42 + 7.70 + 7.348= 8.56 ms

-2

Average of uncertainties = 1 + 1 + 1 + 1 + 2 + 2 + 1 + 28= 1.375= ± 1 ms

-2

= 8.56 ± 1 ms

-2

Percent Error = Theoretical Value – Calculated Value X 100Theoretical Value= 9.81 – 8.56 X 1009.81= 12.7 % Error

Conclusion

Overall, the experiment was carried out with minimum errors and was followedaccording to the given procedure. There were eight data points with three trials each.The investigation was related to the determination of one factor that affects the motionof a simple pendulum followed by the determination of the value of gravity. As it can beseen from the data table and the graph that the value for gravity was calculated to be8.56 ms

-2

± 1 ms

-2

. Furthermore, the percent error was calculated to be 12.7 %.In my experiment, the one factor that affected the motion of the simple pendulum waslength of the string. As the length of the string was decreased, the time period alsodecreased. This is because the velocity increased as the bob had to cover less distancein the given time. Hence, this proves my hypothesis correct

Evaluation

The percent error was calculated to be 12.7 %. This shows that there were someerrors/flaws in my experiment. Firstly, when the pendulum was oscillating, because of itsmomentum, the whole stand was shaking. This irregular movement of the stand affectedthe motion of the pendulum as it did not have the same amplitude throughout henceaffecting the time period. Another error was that when the length of the string wasdecreased, the string was not cut and then fitted but was rounded to the top. Thisaltered the values of my data as it affected the angle of release, which was supposed to

be kept constant. To avoid such mistakes/flaws, the following measures should betaken.

Self Improvement

In order to avoid the above errors/flaws, a number of precaution steps can betaken. Firstly, the string should never be rounded to the top. After each data point, thestring should be removed, measured to the required length using a meter stick and thencut using scissors. This avoids a big error in the experiment. Secondly, it is veryimportant that the stand remains stationary. To do so, extra weight should be put on thebottom but it is also important that this weight is not touching any part of the pendulum.

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A pendulum is any mass which swings back and forth on a rope, string, or chain. Pendulums can be found in old clocks and other machinery. A playground swing is a pendulum.

If you pull the mass away from its rest position, so that the string is at an angle, and then let go, the mass will begin to swing back and forth. The length of time it takes the mass to swing all the way over and back, once, is called the period of the pendulum.

All three experiments will examine things we can do to the pendulum that will change the period. Here are the three questions we are asking:

1. Does the amount of mass on the end of the string affect the period?

2. Does the angle you pull pack the string to affect the period?

3. Does the length of the string affect the period?

In these experiments, the dependent variable will always be the time for one full swing, or the period.

The three tested independent variables will be the mass, the angle, and the length of string.

The controlled variables will be the attachment point of the string, the string itself, the method used to time the pendulum, and the variales we are not currently testing. These will remain the same for each test, so that we know they won't affect the results.

The experiments are easy to do, and don't require any special equipment. We did them ourselves using some string, a few large nuts, a pen, and a watch, and got good results for all three tests in about 20 minutes.

Here'a list of what you'll need for each group doing the experiment:

- a piece of string at least 1 metre long

- 3 or 4 weights, all the same

- a pen and tape, to attach the pendulum to a shelf

- a watch that counts seconds

- pencil and paper to record the results

It's also easier if you have several people doing the experiments, so that one person is free to time the swings.

When you're ready, here are the three experiments. They are set up such that the answers are not apparent, but there arelinks to descriptions of what should happen. There is also a page explaining the pendulum equation, for Science 10 students.

Experiment 1:

Changing the Mass

Experiment 2:

Changing the Angle

Experiment 3:

Changing the Length

For Science 10:

The Pendulum Equation

Variables | Physics | Science & Math | Worsley School

In this experiment, you are going to keep the angle and the string length the same, so they will also be controlled variables.

You will change the amount of mass on the end of the pendulum. This is the independent variable. Then you'll measure the period, which is the dependent variable.

You can print this page to use it while you're doing the experiment.

Changing the Mass

Decide what angle you will use to set the pendulum swinging. Mark it on the wall behind the release point.

Set up the pendulum with a length of string, and tie one weight to the end. Leave enough extra string below the tie point so that you can attach more weights later without changing the length of string above the weight(s).

Trial 1:

Draw the weight back to the release point, and allow it to swing for five full periods (over and back, 5 times).

Time how long it takes to do this.

Then divide your answer by five, to get the time for one full period.

Repeat this twice more. Don't forget to divide by five each time ... we want the time for one period.

Now average your three answers. If you have printed this page, record your answer here:

ONE MASS: Average time for one period: ____________

Trial 2:

Add a weight, so that two are tied to the string. Don't change the length of string above the weight(s).

Draw the weights back to the release point, and allow them to swing for five full periods (over and back, 5 times).

Time how long it takes to do this.

Then divide your answer by five, to get the time for one full period.

Repeat this twice more. Don't forget to divide by five each time ... we want the time for one period.

Now average your three answers. If you have printed this page, record your answer here:

TWO MASSES: Average time for one period: ____________

Trial 3:

Add a weight, so that three are tied to the string. Don't change the length of string above the weight(s).

Draw the weights back to the release point, and allow them to swing for five full periods (over and back, 5 times).

Time how long it takes to do this.

Then divide your answer by five, to get the time for one full period.

Repeat this twice more. Don't forget to divide by five each time ... we want the time for one period.

Now average your three answers. If you have printed this page, record your answer here:

THREE MASSES: Average time for one period: ____________

By now you should have data that will lead you to a conclusion. You might want to write your conclusion here:

When you change the mass on the end of a pendulum,

the period ______________________________

If you would like to see the answer, open this page.

Now try the next experiment.

Experiment 1:

Changing the Mass

Experiment 2:

Changing the Angle

Experiment 3:

Changing the Length

For Science 10:

The Pendulum Equation

In this experiment, you are going to keep the mass and the string length the same, so they will also be controlled variables.

You will change the angle on the pendulum. This is the independent variable. Then you'll measure the period, the dependent variable.

You can print this page to use it while you're doing the experiment.

Changing the Angle

Set up a length of string with one weight on the end.

Trial 1:

Draw the weight back to a steep angle (around 90°) and allow it to swing for five full periods (over and back, 5 times).

Time how long it takes to do this.

Then divide your answer by five, to get the time for one full period.

Repeat this twice more. Don't forget to divide by five each time ... we want the time for one period.

Now average your three answers. If you have printed this page, record your answer here:

BIG ANGLE: Average time for one period: ____________

Trial 2:

Draw the weight back to a less steep angle (around 45°) and allow it to swing for five full periods (over and back, 5 times).

Time how long it takes to do this.

Then divide your answer by five, to get the time for one full period.

Repeat this twice more. Don't forget to divide by five each time ... we want the time for one period.

Now average your three answers. If you have printed this page, record your answer here:

NORMAL ANGLE: Average time for one period: ____________

Trial 3:

Draw the weight back to a small angle (around 20°) and allow it to swing for five full periods (over and back, 5 times).

Time how long it takes to do this.

Then divide your answer by five, to get the time for one full period.

Repeat this twice more. Don't forget to divide by five each time ... we want the time for one period.

Now average your three answers. If you have printed this page, record your answer here:

SMALL ANGLE: Average time for one period: ____________

By now you should have data that will lead you to a conclusion. You might want to write your conclusion here:

In this experiment, you are going to keep the mass and the angle the same, so they will also be controlled variables.

You will change the length of the pendulum. This is the independent variable. Then you'll measure the period, which is the dependent variable.

You can print this page to use it while you're doing the experiment.

Changing the Length

Decide what angle you will use to set the pendulum swinging. Mark it on the wall behind the release point.

Set up the full length of string with one weight on the end.

Trial 1:

Draw the weight back to the marked angle and allow it to swing for five full periods (over and back, 5 times).

Time how long it takes to do this.

Then divide your answer by five, to get the time for one full period.

Repeat this twice more. Don't forget to divide by five each time ... we want the time for one period.

Now average your three answers. If you have printed this page, record your answer here:

LONG STRING: Average time for one period: ____________

Trial 2:

Retie the string at the top so it is about a third shorter.

Draw the weight back to the marked angle and allow it to swing for five full periods (over and back, 5 times).

Time how long it takes to do this.

Then divide your answer by five, to get the time for one full period.

Repeat this twice more. Don't forget to divide by five each time ... we want the time for one period.

Now average your three answers. If you have printed this page, record your answer here:

MEDIUM STRING: Average time for one period: ____________

Trial 3:

Retie the string at the top so it is half as long as the previous trial.

Draw the weight back to the marked angle and allow it to swing for five full periods (over and back, 5 times).

Time how long it takes to do this.

Then divide your answer by five, to get the time for one full period.

Repeat this twice more. Don't forget to divide by five each time ... we want the time for one period.

Now average your three answers. If you have printed this page, record your answer here:

SHORT STRING: Average time for one period: ____________

By now you should have data that will lead you to a conclusion. You might want to write your conclusion here:

When you change the length of a pendulum, the period _______________.

The shorter the string, the ______________ the period.

If you would like to see the answer, open this page.

To see the equation that determines the period of a pendulum, visit this page.

The results of your three experiments were as follows:

• The mass on the end has no effect on the period

• The angle of release has no visible effect on the period

• The length of the string does affect the period. It's a direct variation.

This would lead us to expect that in any equation that will predict the period of a pendulum, there should be a variable for the length of the string, but neither mass nor angle will be involved.

Is there anything else that we didn't test, that might affect how fast a pendulum swings? Something that could make it fall faster, for example? There is, but it's not easy to test with simple apparatus. The force of gravity can affect how fast something falls ... if the force is larger, an object will fall faster.

You may know that the force of gravity is different on other planets, depending on how large and massive they are. On the moon, for example, the gravity is only a sixth of what it is on Earth, so falling objects (and pendulums) would fall more slowly. Similarly, on Jupiter, where the force of gravity is much larger than on Earth, falling objects (and pendulums) would fall much more quickly. You can learn more about the gravity on other planets by trying our Lunar Lander game.

Even on Earth, the force of gravity is not the same everywhere. The farther you are from the centre of the Earth, the smaller the force (and your weight). So you would weigh less while flying (you're high), or more at the North Pole (since the Earth is slightly wider than it is tall). Similarly, if you are standing in a place below which there is a very dense mineral deposit, the force of gravity (and your weight) will be larger. Alternately, if you are standing in a place below which there is a low density pocket, like an oil deposit, the force of gravity (and your weight) will be smaller.

In all cases just mentioned, the difference in your weight is so small as to be unnoticeable. But with very accurate instruments, scientists can measure the strength of gravity at various places on Earth to help determine what's below the ground.

Because the force of gravity will affect how quickly a pendulum falls, there must be a component of the equation allowing for this. A measure of how quickly things fall is the acceleration due to gravity, g, and this is included in the equation as well. An average value for g on Earth is about 9.81 m/s2, but it varies slightly from place to place. The larger its value, the shorter the period, so it's an inverse relationship.

Here's the equation that will allow you to work out the period of a pendulum:

Notice that the period T depends on the length of the pendulum directly. As L gets larger, so will T.

On the other hand, g is on the bottom, so as it gets bigger, the period T will get shorter. If gravity is larger, it falls faster.

This isn't a linear relation, since there is a square root in the equation. But you'll notice that there are no variables for mass or angle, since the period is not affected by changing these.

If you use units m/s2 for g, and metres for length L, then the period units will be seconds

If you square both sides of the equation and rearrange it, you can solve for L:

This will let you calculate the length of a pendulum if you know its period.

To be more accurate, we should mention that the angle of the release does not matter,

as long as the following condition applies:

The angle to which you pull the pendulum mass, and the length of the string, should be values such that the mass's displacement from rest position (s) and its horizontal displacement (d) are of roughly the same size.

For short pendulums, this means the angle should be small.

As long as this is true, the angle will not affect the period, and the equations above can be used.

Other equations describing a pendulum's motion can be derived using potential and kinetic energy relationships.

To learn more about this, go on to the next page.