Pendulum Investigation
Contents
- Plan
- Results
- Analysis and Conclusions
- Evaluation
1. Plan
Aim
To investigate how the length of a simple pendulum affects the time for a complete swing.
Variables
- length
-
The length of the pendulum has a large effect on the time for a complete
swing. As the pendulum gets longer the time increases.
- size of swing
-
Surprisingly, the size of the swing does not have much effect
on the time per swing.
- mass
- The mass of the pendulum also does not affect the time.
- air resistance
- With a small pendulum bob there is very little air resistance. This can easily be seen because it takes a long time for the pendulum to stop swinging, so only a small amount of energy is lost on each swing. A large and light pendulum bob would be affected by a significant amount of air resistance. This might change the way the pendulum moves.
- gravity
- The pendulum is moved by the force of gravity pulling on it. On the Moon, where the pull of gravity is less, I would expect the time for each swing to be longer.
Theory
When the pendulum is at the top of its swing it is momentarily
stationary. It has zero kinetic energy and maximum gravitational
potential energy. As the pendulum falls the potential energy is
transferred to kinetic energy. The speed increases as the
pendulum falls and reaches a maximum at the bottom of the swing.
Here the speed and kinetic energy are a maximum, and the
potential energy is a minimum. As the pendulum rises the kinetic
energy is transferred back to potential energy. The speed of the
pendulum decreases and falls to zero as it reaches the top of its
swing, with the potential energy a maximum again.
[Notes: energy transfers]
A small amount of energy is lost due to air resistance as the pendulum swings. This means each swing is slightly smaller than the one before.
There are two forces acting on the pendulum bob. Gravity tries to pull the bob downwards but this is resisted by the tension in the string. As there are only two forces they can only be balanced when they are in opposite directions. This only occurs when the pendulum is in the middle of its swing, so for the rest of the time the two forces are unbalanced; hence the bob swings back and forth.
The two forces are equal and opposite.
This means there is no resultant force on the bob.
It could either be stationary, or passing through the middle of
the swing.
force due to gravity = weight of bob = mg
m = mass of bob
g = gravitational field strength
g = 10 N/kg at the Earth's surface
The two forces are not in opposite directions. This means
there is a resultant force on the bob to the right.
If the bob is moving to the right the force makes it
accelerate and the speed increases as it moves towards the centre.
If the bob is moving to the left the force decelerates it and it slows down.
[Notes: forces]
Prediction
The diagram shows the arcs through which two pendulums swing. The red one is twice the length of the blue one. Notice that the blue arc is always at a steeper angle than the red arc, and always above it.
The blue pendulum has the most gravitational potential energy at the top of the swing because it is higher. This means the kinetic energy and hence speed through the centre will also be greater than for the red pendulum.
From previous experiments I know that for trolleys running freely down a ramp that the bigger the angle of the ramp the bigger the acceleration of the trolley. This same principle can be applied to the falling pendulums. The steeper the arc the bigger the acceleration of the pendulum will be. A bigger acceleration means a shorter time for each swing. Unlike a ramp the arc of swing is not a straight line. The arc has the steepest gradient at the top and is flat when it reaches the middle. The acceleration of the bob will thus decrease from a maximum at the top of the swing to zero at the centre.
For these reasons as the string gets longer the time per swing
will get longer.
[Notes: prediction limitations]
Method
The cotton is clamped between two small blocks of wood. This
ensures that the cotton swings from a single fixed point. A small
lump of plasticine is attached to the end of the cotton and the
length is adjusted by pulling the cotton through the two blocks.
Gravity may be considered to act through the centre of gravity of
the bob. For this reason the length of the cotton is measured
from the wooden blocks to the centre of the bob.
Timings for twenty complete swings are started and stopped as the pendulum passes through the mid-point. A long pin is set up at the mid-point, at right angles to the plane of swing, to provide an accurate reference point. This is achieved by positioning yourself so that you are looking directly along the line of the pin. As the cotton passes the point the stopwatch is started and counting is started at "0". The pendulum will swing to one side, then back through the centre and to the other side. When it passes the centre again "1" is counted for the first complete swing. This is repeated and on the last swing you must again look along the pin and the timer is stopped as the cotton passes.
Each timing is repeated three times. It is important to check
that the pendulum is swinging in a single plane before
measurements are started. The size of swing should also be kept
small.
[Notes: Accuracy of the method]
Equipment List
cotton, plasticine, metre rulers, digital stopwatch, long pin
Risk Assessment
There are no safety risks involved.
| Dependent Variable | value | how measured |
|---|---|---|
| Time for one complete swing (Period) |
time for 20 swings. 3 repeats for each length |
digital stopwatch |
| Independent Variable | ||
| length | 20 to 160cm in 20cm steps | ruler |
| Control Variables | ||
| size of swing | small (10° or less) | protractor |
| mass | 10g | electronic balance |
| air resistance | very small | |
| gravity | 10 N/kg | |
Trial Data
1. Varying mass of bob
The mass was changed by a factor of five and this had little effect on the time.
| length (cm) | mass (g) |
displacement (cm) | time (20 swings) (s) |
|---|---|---|---|
| 58 | 5 | 10 | 30.63 |
| 58 | 25 | 10 | 30.75 |
2. Varying displacement of swing
The size of the swing was changed by a factor of three and this had little effect on the time.
| length (cm) | mass (g) |
displacement (cm) | time (20 swings) (s) |
|---|---|---|---|
| 58 | 25 | 10 | 30.94 |
| 58 | 25 | 30 | 31.33 |
3. Varying length
The length was changed by a factor of two. The time increased as the length increased but by a factor of 1.4 approximately.
| length (cm) | mass (g) |
displacement (cm) | time (20 swings) (s) |
|---|---|---|---|
| 30 | 25 | 10 | 22.37 |
| 58 | 25 | 10 | 30.75 |
From the trial data it is easily seen that the length of the pendulum is the only variable that has a significant effect on the time of swing.
2. Results
| length (cm) |
√length (√cm) |
number of swings N |
time for N swings (s) |
average time N swings (s) |
average time 1 swing (s) |
||
|---|---|---|---|---|---|---|---|
| 5.0 | 2.24 | 50 | 22.84 | 22.91 | 22.88 | 22.88 | 0.46 |
| 10.0 | 3.16 | 50 | 32.16 | 32.25 | 32.19 | 32.20 | 0.64 |
| 20.0 | 4.47 | 20 | 18.10 | 18.06 | 18.03 | 18.06 | 0.90 |
| 40.0 | 6.32 | 20 | 25.40 | 25.47 | 25.34 | 25.40 | 1.27 |
| 60.0 | 7.75 | 20 | 31.09 | 31.08 | 31.06 | 31.08 | 1.55 |
| 80.0 | 8.94 | 20 | 35.94 | 35.90 | 35.91 | 35.92 | 1.80 |
| 100.0 | 10.00 | 20 | 40.12 | 40.08 | 40.09 | 40.10 | 2.00 |
| 120.0 | 10.95 | 20 | 43.97 | 43.97 | 43.90 | 43.95 | 2.20 |
| 140.0 | 11.83 | 20 | 47.31 | 47.37 | 47.44 | 47.37 | 2.37 |
| 160.0 | 12.65 | 20 | 50.72 | 50.75 | 50.75 | 50.74 | 2.54 |
Control variables
mass of bob = 10g
size of each swing kept small (maximum displacement about 10cm)
Changes to the plan
Extra results were taken at lengths of 5cm and 10cm. This was done to help extend
the range over which the trend in the data could be tested.
The number of swings timed was increased from 20 to 50 for these two lengths.
This prevented a possible drop in the accuracy if the total recorded time was to become
too short.
[Notes: results]
3. Analysis and Conclusions
Graph of results
(Remember, in a proper writeup the graphs would be printed out A4 sized, not this small)
Graph1:
To show how the time per swing varies with the length
Graph1 shows that the time for each swing increases as the length increases.
The gradient of the graph decreases as the length increases. This shows that the rate of increase of time per swing decreases as the length increases.
Graph2:
To show how the time per swing varies with the square root of the length
Graph2 shows the time for each swing plotted against the square-root of the length. This gives a straight line graph through the origin.
The equation for a straight line through the origin is;
y = mx
The gradient m measured from the graph = 2.5÷12.5 = 0.20
If T is the time for one swing in seconds, and L is the length in centimetres, the equation for the line can be written as;
T = 0.20√L
Conclusions
The time for one complete swing is proportional to the square root of the length. All the points for Graph2 lie on a straight line so the conclusion is very reliable over this range. It seems likely that the same trend would continue if the string was made longer. Shorter lengths look like they would also follow the same pattern although it gets more difficult to take the measurements as the time gets shorter. For very short lengths the trend may not continue and would be very difficult to measure. It is not possible to try lengths shorter than the diameter of the bob, for instance. A much smaller bob on a very fine and lightweight filament could be made and tested.
I was able to predict that the time would increase but can not prove that it is proportional to the square root of the length.
4. Evaluation
Accuracy of Results
Measuring the length
One difficulty when measuring the length is deciding where the centre of the bob is. The uncertainty in determining this measurement is probably about 1 mm. Measuring the length beyond about 1 metre is more difficult than short lengths because the measurement has to be done in two parts using metre rulers. The total error in measuring the longest length (160 cm) could be 2 or possibly 3 millimetres. For the shortest length the error is not likely to be more than a millimetre.
Adjusting the length of the pendulum to an 'exact' value such as 40.0mm was not as difficult as I thought it would be. The cotton could quite easily be slowly pulled through the wooden blocks a little bit at a time until the measured length was correct. Had this not been possible, or to save a little bit of time, any value close to the required value would have been perfectly acceptable and would not have made any difference to the conclusions.
Measuring the time
The stopwatch measures to one hundredth of a second although the overall accuracy of the time measurements are not that good. The human reaction time to start and stop the watch roughly cancel each other out as the same event is being observed, and reacted to in the same way, each time. Errors are produced by any variability in the reaction time of the individual which could be affected by many things. The following table gives the difference between the maximum and minimum values of time for each length.
| length (cm) | 5 | 10 | 20 | 40 | 60 | 80 | 100 | 120 | 140 | 160 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| time range (s) | .07 | .09 | .07 | .13 | .03 | .04 | .04 | .07 | .13 | .03 | avg = .07 |
The maximum range is 0.13 seconds and the average is .07 seconds. This shows that the timings are repeatable to within a couple of tenths of a second. Taking more time measurements may give a slightly more accurate average for each length, but not by much.
The smooth trend in the graph indicates that the results are accurate and reliable. There are no anomalous results or anomalies to be seen in the trend of the graph.
Reliability
No significant problems or difficulties were encountered when carrying out this investigation. The accuracy and reliability of the results and conclusions are very good. Within the accuracy of the method used, and for the range of values investigated, it is clear that the time for a complete swing of the pendulum is proportional to the square root of the length.
Improvements
The procedure used was simple and straightforward and no
difficulties were encountered.
A small improvement could be made
to measuring the length of the pendulum. A longer rule, or piece
of wood, could be placed level with the point of suspension, and
a set square could be placed along the flat side and just
touching the bottom of the pendulum. This distance could then be
measured more accurately than trying to guess where the middle of
the bob is. The diameter of the bob could be accurately measured
with some vernier callipers so that the true length of the
pendulum could then be calculated.
The thread used was quite stretchy. If the investigation was repeated I would replace it with something more rigid, such as extra strong button thread.
More repeats could be taken but I don't think this would add much to the accuracy of the conclusions.
Longer lengths could easily be tried, up to whatever maximum could be obtained. With a suitable location a length of several metres could be obtained. If the pendulum gets too long then stronger thread and a heavier bob might be needed. It would be interesting to try shorter and shorter lengths although a limit would be reached when the pendulum moves too quickly to be accurately counted. It may be possible to have some sort of electronic detection system that could automatically count and time the swings. Something like a light gate as part of a computer based logging system might work. An alternative might be a very high speed digital video camera that could accurately record the position of the bob and the elapsed time.
Extending the investigation
Suggestions have already been made to extend the range of
lengths investigated and to see if the observed trend continues.
There is an alternative way that a pendulum can swing.
Instead of swinging backwards and forwards in a single plane it
is possible to make the pendulum swing in a horizontal circular
path. It would be interesting to investigate how the time for
each revolution of this conical pendulum changes with the
length and to compare this with the ordinary simple pendulum.