EXPERIMENTAL PROJECT 2012-2013
Rigid Pendulum Experiment
INTRODUCTION
The simple pendulum is familiar idea to many students as they will have seen a small mass
swinging from side to side at the end of a long piece of string. For such a simple piece of
apparatus, it has a remarkably distinguished history; from Galileo to Huygens who, in the
seventeenth century, developed the pendulum clock. This used the excellent timing
properties of the pendulum to set the rate of a clock which could even have a seconds hand
on the face. As long as the amplitude remained quite small, the period of the pendulum (the
time for it to swing from one side to the other and then back again) was not dependent upon
the amplitude of the swing.
Measuring g
With the development of astronomical observations for timing and better clocks, particularly
in the nineteenth century, instead of using the pendulum as a clock itself, the period of the
pendulum along with its length could be measured to determine a value for the acceleration
due to gravity, g, at the surface of the earth. In order to measure the value of g very
accurately at different places around the world, the length of the pendulum needed to be
known accurately. For a simple pendulum, this is surprisingly difficult. In this age of scientific
endeavour at the end of the eighteenth century, a pendulum was developed that was
constructed of a rigid rod with knife edges at each end, so that the pendulum could be timed
and then suspended upside down and timed again. Adjustments could be made to the
supports and then the distance between the knife edges could be made with great accuracy.
This gave astonishingly precise values for g. For this, the pendulum had to be a metal rod
rather than a light string and so the mass was no longer concentrated at one end. The
experiments below form a much simplified version of this approach, in which we change the
length of the pendulum rod and then see how the period changes as a result. We will see how
a straight line graph can be obtained, even for a complicated equation in which we cannot
separate out the two variables to be plotted on to different sides of the equation. A value for
g can be obtained, even though it would appear to be rather well hidden in the theory.
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APPARATUS FOR EXPERIMENT
1 One retort stand consisting of a metal base, a metal rod, a metal clamp and a metal
boss. The boss enables the clamp to be attached to the rod.
2 A second retort stand rod and boss are required to make the pendulum. The rod
should have a much greater mass than the boss, and so an iron rod and a light alloy
boss would be ideal (an iron boss would be suitable also).
3 A knife edge; for example a blunt kitchen knife, the smooth rear edge of a hacksaw
blade or any suitable object with a knife edge. This should not be so sharp that
catching your finger on it will cause an injury. It must be rigid enough so that it does
not bend back and forth as the pendulum rod swings.
4 A means of timing the oscillations of a pendulum (a stopwatch).
EXPERIMENTAL SET-UP
The first retort stand should be set up with the clamp attached to the rod by the boss. The
knife edge should be held firmly in the clamp with the knife edge horizontal, and facing
vertically upwards, Figure 1.
Figure 1
Figure 2
The pendulum is made by attaching the second boss to the rod, which is then balanced on the
knife edge at a fixed point on the boss, so that the rod can oscillate in a vertical plane
perpendicular to the knife edge; the rod will not be vertical when it is in equilibrium, Figure 2.
If the thread of the boss screw is used as a pivot on the blade, then the oscillations die away
too quickly. Either of the arrangements shown in Figures 3 & 4 could be used, although it is
preferable to have the pivot point as close as possible to the suspended rod, as in Figure 3
rather than Figure 4. It should be possible to support the boss as in Figure 5, very close to the
rod, and this is the ideal position of the pivot.
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The oscillating rod and its boss supported on a hacksaw blade.
Figure 3 Suitable pivot
Figure 4 Poor pivot position
Figure 5 Good pivot position
The distance between the knife edge support and the centre of mass of the rod is required,
that is distance L which is shown in Figure 6. Firstly, we can assume that the centre of mass of
the rod remains fixed (there is only a slight variation as we move the position of the boss) and
that the centre of mass lies along the long axis of the rod.
Secondly, if the pivot point on the boss is close to the rod (1 – 2 cm say) as shown in Figure 5,
then the distance L is to a good approximation equal to the length, x along the rod itself and
this makes for simpler measurements. The position of the centre of gravity can be obtained
by balancing the rod and its boss horizontally on the knife edge, marking the position with a
pencil, and determining either x or L as appropriate.
L
x
Figure 6
Before you begin the experiment itself, you should look at the Report Guidelines attached. Rather
than just obtain a set of numbers, you should make sure that you can see what is happening to the
pendulum as you change the length x. Can you predict what the first graph will look like? Plot the
first graph before you put the apparatus away. Make sure that you are able to comment on the
issues referred to in the Guidelines; read the Guidelines now.
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EXPERIMENT: PART A
Measure the time period, T, of oscillations of the rod and boss for small amplitude oscillations
about its equilibrium position. This should be performed for different settings of the position
of the boss along the rod. In each case either x or L, as appropriate, should be measured.
Draw up a table of values of T and x or L, giving an uncertainty and the units. Then:
1. Plot a graph of T against x or L. [From now on only x will be referred to]
2. Determine the minimum value of T, which we call TMIN , and the associated value of x,
which we call xMIN, estimating the uncertainty on the values obtained. It may be
necessary to take additional measurements around (TMIN, xMIN) to obtain accurate
values of these quantities.
3. Plot a graph of xT2 against x2.
4. Assuming a linear relation of the form
xT2 = A + Bx2,
where A and B are constants , determine A and B. (If the approximations made are
good ones then B is equal to 4π2/g).
EXPERIMENT: PART B
In this experiment a rod, from a further retort stand, plus boss are permanently attached to
the bottom of the ‘oscillating rod’ from the first experiment. The length of rod on one side of
the boss should be adjusted so that when the new system is suspended, in equilibrium, from
the knife edge, the central rod is vertical and the added rod is horizontal, Figure 7. It is to
oscillate in a vertical plane, perpendicular to the knife edge.
1. Repeat the measurements of T and x (or L) performed in the first experiment.
2. Carry out the Instructions 1 – 4 of the first experiment.
Figure 7
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SUMMARY AND WRITE UP
Once you have finished Part A and Part B of the experiment complete a summary table. The
format of the table is given below.
Then write up the experiment in a report. See the Report Guidelines for writing the
experimental report. You do not need to write two reports – just write comments about the
changes you made for Part B and comment on what makes it a slightly different experiment.
SUMMARY TABLE
PART A
PART B
TMIN
xMIN
A
B
(A/B)
½
(4AB)
¼
Remember to include the units for all the quantities in the table.
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REPORT GUIDELINES
You should give a full but concise description of the experimental procedure (you do not need
to repeat the instructions given here). Highlight any particular ideas you had to make the
results more reliable, with results, tables, units, uncertainties, graphs, diagrams and images.
This is elaborated below.
•
This set of instructions gives the practical details of the setup so you do not need to
describe them. You must write about any additional points, the particular apparatus that
you used (iron rod, aluminium boss, etc.), precautions you took, good ideas you had about
making the results more precise, what were the difficult measurements to make and why,
whether your pivot was close enough to the rod so that you could use x rather than L,
how good the approximation was (consider Pythagoras); does it make a difference if you
swing the rod through two or three times the angle of swing instead – show how you
checked this, not just saying that you did.
•
You should describe briefly in words the behaviour of the rod when the boss is closer to
the middle instead of at one end; this is so that you get to understand how the apparatus
behaves before you start taking readings.
•
Take a good range of data, and plot the graphs before you put the apparatus away. You
may find that you do not want an even spread of results, but that you need more at some
particular values of x and rather fewer at other values of x. You do not have to take all of
the readings in the right order. They will sort themselves out on the graph.
•
Results tables should have the original data including any repeated measurements, with
the units at the top of the column along with the correct symbol for the quantity. A
column for the average can then be worked out. Can you include a column with an
estimate of the uncertainty on your average value?
•
If you feel that the results are not right, then don’t scribble them out or delete them; keep
them and just point out what might be wrong with them or why you have no confidence
in them. The purpose is to show that you have investigated the apparatus and given it a
good try out so that you can convince the reader that you know what you are talking
about. Things can go wrong and you need to show how you can overcome setbacks. If you
delete everything that does not seem right to you, the reader might see a nice final table
of results, but he does not know how much investigative effort you have put in to get that
set of results. The reader might indeed wonder whether they are that good. Do not write
an essay, just a few bullet point comments.
•
Graphs should have labelled axes, units on the axes, a whole sheet of graph paper, some
gridlines if done in Excel so that the reader can get values off the graph, a scatter graph
with a line of best fit (trendline in Excel) – definitely do not “join the points”, display the
equation for the trendline on the graph. Comment on the graph i.e. look at it and state
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what you can see about the shape of the graph, the scattering of the points, does it go
anywhere near the origin, is it a straight line or a curve (or is it hard to tell), is the data
close enough to the line of best fit to agree with the uncertainties you wrote down in your
table of results?
•
The conclusion is the table of results that you fill in, but you should also make at least a
comment about whether you yourself think that the results are believable or not, and
why.
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