Download 4. the simple pendulum

4. THE SIMPLE PENDULUM
By means of an empirical method you will deduce the relationship between the period of
the simple pendulum, the length of the string, the mass of the object, and the angle the
object is initially displaced.
Why Study the Simple Pendulum?
The simple pendulum is an ideal system on which
to apply the empirical method, a method which
embodies the very essence of science.
The empirical method enables one to deduce a
mathematical relationship which describes a physical effect when a theory is unavailable. This is
likely to happen most often for new discoveries.
The empirical method begins with exploratory
experimentation, follows with deduction based on
the results, then more experimentation, and so on,
until the description sought for is found. Now a
theory does exist for the simple pendulum, so the
Theory Section that would ordinarily be placed at
this point in the guidesheet is placed near the end;
you are expected to pretend that the theory does
not exist so you can experience the empirical
approach for yourself. This was the approach used
by Galileo more than 400 years ago on this very
same problem. You are on your honor not to look
at the theory until the proper time!
The Pendulum and Its Period
A simple pendulum consists of an object of point
mass1 m suspended on the end of a string of length
l as sketched in Figures 1 and 2. When the object is
displaced from its equilibrium position it swings
back and forth with a well defined motion. The
period of the motion is, by definition, the time for
one complete oscillation—the time for the object to
move from position A (Figure 1) to position B and
back to position A again. It is natural to hypothesize that the period T of the pendulum depends in
some way on the mass m, the length l and the
angle θ the string is displaced from its equilibrium
(vertical) position. It is the object of this experiment to uncover this relationship.
knot
θ
toothpick wedge
l
string
B
O
m
A
T: Time for one oscillation
Figure 1. The period of a simple pendulum depends in
some way on m, l, and θ.
Figure 2. A way of supporting a simple
pendulum to constrain its motion to a plane.
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4 The Simple Pendulum
The Experiment
Exercise 0. A Practice Run
Orientation
Identify the apparatus: three metal cylindrical
objects of different mass, a piece of string, a stand,
a meter rule, a vernier caliper, a Prosonic model
1301 stopwatch (precision ±0.01s), a few toothpicks
and a protractor for measuring angles.
First Run
Most experiments in science benefit from practice
runs to enable the experimenter to become
familiarized with the equipment and to iron out
bugs in procedures. This exercise is a practice run.
Choose any one of the cylindrical objects and
some arbitrary length of string. If you wish, set up
the pendulum as shown in Figure 2. Measure the
length carefully, keeping in mind that l is the distance from the point of suspension to the center of
mass of the object. (Estimating the position of the
center of mass will involve some error; this is
natural.) Set the object swinging with a small amplitude, measure the time for some convenient num-
ber of oscillations (say 10) and from this total time
Ttotal calculate the period T, the time for one oscillation.2 Ordinarily you would measure only one
Ttotal for each combination of variables. But first we
must find the uncertainty in T.
Finding the Error in T
How does one go about finding the error in T? One
approach is to measure T a number of times, to
form a sample, and then take one standard deviation of this sample as the error in a single measurement. The idea is illustrated in Figure 3. (Recall
this subject was covered in Exercise 3 of the
Orientation Workshop.) Of course, the result must
be rounded, as usual, to one signaificant digit—
here 0.02 s. Do this now for the mass and length
you chose before. This standard deviation you can
now use as the error in every subsequent measurement of T.
Period (s)
1.43
1.41
1.44
1.42
1.43
1.41
1.44
1.45
1.48
1.4
Standard Deviation>> 0.023309512
Figure3. If the period is measured several times to form a sample, the standard deviation of the sample can be
calculated (as here on an Excel spreadsheet). The standard deviation can then be inferred as the error in all
subsequent measurements of the period.
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The Simple Pendulum 4
Exercise 1. Collecting the Data
The most logical way to test for dependence when more than two variables are involved
is to change only one variable at a time keeping the others constant. This is the strategy of
this exercise.
Mass Dependence
Vary m for a given l and θ and calculate T each
time using the procedure of Exercise 0. The three
metal objects may be used singly or together in
groups. Collect as many (mi, Ti) datapairs as you
can. From casual observation does T seem to
depend on m? The proper analysis you will carry
out in Exercise 2.
Length Dependence
Vary l for a given m and θ and calculate T each
time, again using the same procedure as before.
Collect as many (li, Ti) datapairs as you can by
varying l over as wide a range as you can. From
casual observation does the data seem to indicate a
dependence of T on length?
Angle Dependence (Optional)
Choose fixed values of m and l and vary θ from 10º
to about 90º and find T each time. From casual
obervation does T seem to depend on θ?
Exercise 2. Analyzing the Data
Mass Dependence
Though your data most likely does not appear to tion, i.e., a dependence probably exists. If R is very
indicate a dependence on mass the definitive test is small then a correlation probably does not exist.
to plot the (mi, Ti) data in a program like pro Fit , fit Do this now. The more datapairs you collect the
a line to the data and examine the correlation coef- more likely this test will work.3 An example graph
ficient R. If R is very nearly unity then a correla- output from pro Fit is shown in Figures 4 and 5.
Study of Mass Dependence
Period (sec)
1.20
1.10
1.00
0.04
0.08
0.12
0.16
Mass (kg)
Figure 4. The graph from proFit for a typical T vs m dataset. The fit shows a sraight line of very small slope,
indicating a possible non-dependence. The definitive test is a small correlation coefficient—see Figure 5.
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4 The Simple Pendulum
Figure 5. The results of the fit whose graph is shown in Figure 4. The correlation coefficient is 0.64 indicating
strongly that there is likely no dependence of T on m.
Length Dependence
Your data has probably indicated to you that T
depends somehow on l. One good guess as to this
dependence is the general function
T = kl n
…[1]
where, of course, T and l are variables and k and n
are—as yet unknown—constants.
Now eq[1] looks rather complicated. This function can be simplified by the technique of linearization—here by taking the natural logarithm of
both sides. The result is
ln(T) = ln(k) + n ln(l) ,
…[2]
which is now in the form of a straight line
Y = b + mX
with the following transformation of variables:
Y → ln (T)
X → ln (l)
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and where
and
b (the intercept) = ln (k)
m (the slope) = n.
Therefore go ahead and enter your (l i, T i) data into
pro Fit, make the transformation and fit to a
straight line. From the fit results find the values of
k and n and their errors. An example of this
activity is illustrated in Figures 6 and 7.
You should now be able to state that the
relationship
T = ( k ± ∆k )l ( n±∆ n)
…[3]
describes your data …or not. In Exercise 3 you can
compare your results with the theory.
Angle Dependence
Question:
? How would you go about studying the dependence of T on θ?
2. The Simple Pendulum
Ln(T) vs Ln(l) for Simple Pendulum
Ln (T)
0.5
0.0
-2.0
-1.5
-1.0
-0.5
0.0
Ln (l)
Figure 6. A straight line fit to a typical set of pendulum data of T vs l.
Figure 7. An example of the results window of proFit for a typical dataset for a simple pendulum. The correlation
coefficient is 0.999 indicating strongly a dependence of T on l.
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4 The Simple Pendulum
Exercise 3. Comparing Your Results With Theory
Of course, a theory does exist to explain the motion of a simple pendulum. Read the theory below
if you have not done so already. A relationship of
the form of eq[1] is, indeed, predicted by the
theory—that is, eq[7b] below is of the form of
eq[1].
Questions:
? What values for k and n does the theory,
eq[7b], predict? (Take g = 9.804 ms–2.)
? Do your empirical values of k and n agree with
the theoretical values to within your experimental error? What do you conclude about the
usefulness of the empirical method in this
case?
Addendum. Theory of the Simple Pendulum
We assume that the simple pendulum (Figure 8)
consists of an object of point mass m on the end of
a string of length l suspended from a rigid point P.
We assume also that the string is of negligible
mass. The resultant force F which causes the object
to swing back and forth through the equilibrium
position O is the component of the force of gravity
m g acting along a tangent to the object’s circular
path. Two forces act on the object as is shown in
the freebody diagram in Figure 9.
P
θ
l
T (tension
in string)
m
m
F
m g
m g
Figure 8. A simple pendulum showing the force
of gravity m g and its component F.
Figure 9. The freebody diagram of the object
with two forces acting on it.
The component of the force of gravity acting along
a tangent to the object’s path is
r
F = F = mg sinθ .
Since F is a restoring force we give it a minus sign
and use Newton’s Second Law of Motion to write:
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θ
F = –ma = mgsinθ ,
so that
a = –gsinθ .
Using a little calculus the angular acceleration α of
the system is given by 4
The Simple Pendulum 4
d dθ d 2θ
= 2 ,
dt dt
dt
α=
Eq[5] can be integrated to find a solution for θ or a
solution can be guessed at. You should be able to
show that a solution is 5
and since the linear and angular accelerations are
related by a = lα we can write
α= l
so that
d 2θ
= –gsinθ ,
dt2
d 2θ g
+ sinθ = 0 .
dt 2 l
…[6]
where θo is the amplitude and f is the frequency in
hertz (Hz). Substituting eq[6] into eq[5] satisfies
the equation provided
…[4]
This is a differential equation of the second order.
Its solution is very difficult to obtain. However, if θ
is “small enough” we can make the following
approximation
f =
1
2π
Eq[4] then reduces to
…[5]
g
.
l
…[7a]
Therefore, for a simple pendulum where θ is
always “small”,
period = T =
sinθ ≈ θ in radians .
d 2θ g
+ θ = 0.
dt 2 l
θ =θ 0 sin(2πft) ,
1
l
= 2π
.
f
g
…[7b]
Note that the formula predicts that T is independent
of m and independent of θ provided θ is “small
enough”.
Videos and Physics Demonstrations on LaserDisc
The Vision of Galileo, with David Suzuki, Tape #16
from Chapter 3 Linear Dynamics
Demo 03-13 Galileo’s Pendulum
Demo 03-14 Bowling Ball Pendulum
Activities Using Maple
E04The Simple Pendulum
In this worksheet three kinds of pendulums are examined: The Simple Pendulum, The Non-Linear
Pendulum and The Damped Simple Pendulum. For a demo of aspects of the period of pendulums see the
Maple worksheet Period of a Pendulum.
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4 The Simple Pendulum
J Perz
Stuart Quick 94
EndNotes for The Simple Pendulum
1
By referring to the object as a point mass we are assuming the mass is concentrated in a point of negligible size.
This way we avoid having to deal with the object’s internal energy.
2
This term has a precise meaning in physics. A period is not just some duration of time as it is in everyday usage;
in physics it is the time taken for one complete oscillation.
3
A dependence probably exists if R is of the order 1, or conversely probably does not exist if R is about 0.7 or less
(this number is somewhat arbitrary). This procedure may fail if the sample is small.
4
You will not be held responsible for the calculus here. The following analogies may be of help to you:
linear displacement x → angular displacement θ, linear velocity v → angular velocity ω
linear acceleration a → angular acceleration α. Also,
2
a = dv = d x
dt
dt 2
You may find these expressions useful: dθ
dt
v = dx
dt
5
ω = dθ
dt
2
α = dω = d θ .
dt
dt 2
= 2π f θo cos 2π ft
2
and d θ = – 4π2 f 2 θo sin 2π ft .
dt 2
When substituted into eq[5] these expressions will satisfy that equation provided the relationship eq[7a] holds.
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