The SIMPLE PENDULUM SIMPLE HARMONIC OSCILLATOR I

GOAL: To experimentally determine which
characteristics of a pendulum influence its period. To
determine functional relationship between a period’s
period and its characteristics.
INTRODUCTION: Consider a simple pendulum
consisting of a spherical object or “bob” suspended by
a string. How would one describe such a pendulum?
What characteristics of this pendulum can one
measure? One could measure the mass of the bob, the
diameter of the bob, the color of the bob, the length of
the string, the diameter of the string, the mass of the
string, the color of the string, and probably lots of
other things. When the pendulum is in motion, one
could measure the maximum angle to which the
pendulum swings.
The period, T, of a pendulum is the time it
takes for the pendulum to complete one cycle, i.e., to
return to the same position with the same velocity. For
example, as the pendulum passes a point in its motion,
swinging from right to left, start timing. Next, the
pendulum passes the same point now swinging from
left to right, but its velocity is the negative of the
initial velocity. Finally, the pendulum passes the point
again but this time swinging right to left so that it has
the original velocity in both magnitude and direction,
thus the cycle is completed and the timing of the
period stops.
Intuitively, what characteristics of the
pendulum might affect its period. Color probably does
not really change the system mechanically. If the
diameter of the string is relatively small, this is
probably not important. If the mass of the string is
much smaller than the mass of the bob, then the mass
of the string probably is not important.
But the mass, m, of the bob enters into the
equation of motion ( ) and may be important.
Test #1: With all other things being equal, see if the
mass of the bob effects the period.
What about the diameter of the bob? If the
diameter of the bob is small compared to the length of
the string, then one might expect that one could treat
The SIMPLE PENDULUM
SIMPLE HARMONIC OSCILLATOR I
Physics 141
Pendulum -- 1
the bob as a point mass located at the center of the
bob.
Then the radius of the bob would have to be added to
the length of the string to obtain an effective length of
the pendulum. This length helps to define angles that
are important in the resolving forces so it could be
important.
Test #2 is to see if varying the effective length ( + r)
of the pendulum effects its period.
Figure 1 The pendulum.
There is the maximum angle, max that the bob
swings through on each cycle. The larger the angle the
greater the distance the bob travels for a given
pendulum length so this probably has some effect on
the period.
Test #3 is to vary the maximum angle of the pendulum
and see if the period varies.
What happens if because of something odd in
your release, the bob does travel back and forth along
an arc but rather travels an oval path? Does this effect
the period?
Test #4 is to try oval paths.
Try to determine the mathematical relationship
between the period and any of these four variables the
effect the period.

period.
PROCEDURE:
Initially, use the stop watch for measuring the
Make Test #1, 2, & 3. It may be to your
advantage to initially make these tests quickly, because
there may be ranges for these parameters where the
period varies rapidly. You may wish to avoid these
regions when testing the other parameters.
When working with small angles (less than
10°) and smaller balls, you may wish to use the
photogates. The suggested settings are: PENDulum
mode., MEMory ON, and either 1 ms or 0.1 ms. DO
NOT HIT THE PHOTOGATE WITH THE BOB!
After general behavior of the period as a
function of each parameter is known, then make
detailed studies of the primary causes of variation.
Use the nature of the linear and maybe a log-log plot to
determine the functional relationship.
Take enough data to make a reasonable
estimate of your errors.
Compare your results to eqs. 6 and 9.
Determine the gravitational constant, g, using
eq.10.
WARNINGS:
DO NOT USE the photogate and timer when
taking preliminary data, working at angles greater than
10° or using heavy bobs. This is to avoid the problem
of the bob hitting and damaging the photogate.
Before using the photogate, practice to obtain
a clean release of the bob, such that the path of the
pendulum is in a plane rather than an ellipse.
Pendulum stands may be at face and eye level,
please be careful. Do not hit yourself or anyone else
with the pendulum.
EQUIPMENT:
Long vertical support rod
Pendulum clamping bar
Table & rod clamps
Stopwatch
Timer & photogate
Meter & ruler
In room
Balance ( triple beam)
Various pendulum bobs
String & scissors
Physics 141
Pendulum -- 2
Equipment notes:
When using a stopwatch to measure the period
of the pendulum, measuring more than one (maybe
ten) cycles and then dividing by the number of cycles
may help reduce errors associated with your reflexes.
When using the stopwatch for timing,
Be careful not to count the initial half
cycle as a full cycle.
As the pendulum swings through the
bottom of its swing and you start the
stopwatch, be careful to start counting
at zero and not one.
Be careful that the string is clamped equally
on both sides. Otherwise the length of the pendulum
may be not longer on one side of the swing than the
other.
ANALYSIS:
A i. What are the uncertainties in timing the period
for each different method used to measure it? How
does reducing the uncertainties in measuring the
period affect the uncertainty in the gravitational
constant?
A ii. Compare the quality of timing results for using
the stopwatch and using the photogate and timer. The
uncertainty in timing with the photogate is probably
100 times smaller than for using the stopwatch. Does
your data indicate that this is true? Discuss.
A iii. Consider the uncertainty in timing one period
associated with the human reflex using a stopwatch. If
the combined uncertainty for both starting and
stopping the stopwatch is 0.2 seconds and one is
timing a 0.5 second period, what is the relative
uncertainty? If one times ten full cycles and then
divides by ten, what is the uncertainty in one period
caused by the 0.2 second timing uncertainty associated
with the human reflex? Would timing one hundred
cycles be even better? Why?
A iv. How good is the approximation that sin() =
for the range of angles you measured? Compare sin()
and for 0°, 1°, 2°, 5°, 10°, 15°, 20°, and 25°. From
your data, what is the smallest angle that you can see
deviation from the expected independence of period on
maximum angle of swing.

THEORY I:
Consider the free body diagram of the
pendulum bob treated as a point mass, suspended by a
massless string of zero diameter):
Figure 2 The free body diagram for the
bob. The resolution of the weight, w into
components w and w is shown.
x y
There is a radial acceleration parallel to the
length of the string (x direction), so the sum of the
forces along the x axis must be such that:
(1)
Perpendicular to the string the only force is w y,
thus
(2)
Next consider the torque about the point where the
string is clamped. The distance from this point to the
center of the bob is eff = + r, so the torque, , and
moment of inertia, I, about this pivot point are:
(3)
Analysis of this problem begins with the rotational
form of Newton’s second law:
Physics 141
Pendulum -- 3
(4)
For small angles, sin() = , if one is using radians
rather than degrees.
(5)
This equation has the general form of the
simple harmonic oscillator equation, eq. 15, and has a
solution:
(6)
The period, P, is the time interval required for
sin(t + ) to complete one cycle, that is:
Or:
0
(7)
(8)
Combine eq. 6 and 8 into an equation that expresses
some power of the period, P, as a linear function of the
effective length of the pendulum, eff.
where n is some number.
(9)
Q i. Find the gravitational constant, g as a function
of the slope, C in eq. 9:
(10)

THEORY II
General Simple Harmonic Oscillator Problem
The simple harmonic oscillator is one of the
central problems in physics. Many mechanic systems
demonstrate simple harmonic motion. The same
simple harmonic oscillation are involved in a large
number of different circumstances including: sound,
light, electrical circuits (such as LRC circuits that
consists of an inductor, a capacitor, and a resistors in a
loop), crystal lattice interactions, and quantum
mechanical systems. Some systems that exhibit simple
harmonic motion can be driven into chaotic motion so
understanding the simpler motion is the first step in
understanding the complexities of chaos.
For mechanical systems, the theoretical
understanding of simple harmonic motion begins with:
and in one dimension this is:
(11)
(12)
In simple harmonic oscillator systems, the force, F,
opposes the displacement but is proportional to the
displacement, x, with the zero of the x position
selected such that the force is zero at x = 0:
or:
Eq. 2 becomes:
(13)
(14)
(15)
which is a thing called a differential equation, and has
a well known solution. Most of the time one solves a
differential equation by remembering some function
that behaves something like what you need,
substituting this possible solution into the differential
equation, and solving for any free parameters
Physics 141
Pendulum -- 4
(repeating the guessing game, if necessary).
Recall that:
Thus:
(16)
(17)
Note the similarity in form between eq. 15 and eq. 17.
(Because sin( + /2 ) = cos(), the two equations in
eq.17 are in fact equivalent.) Eq.15 and 17 are the
same if in eq. 15 one makes the following assignments:
(18)
where A is the maximum displacement. Eq. 19 is the
solution for ALL simple harmonic oscillator problems.
Usually one selects the zero point for the time, t, such
that 0 = 0. All that remains is to know the value of
the constant, C.
Using eq. 18 with 0 = 0,as the solution to
eq. 15 consider:
Q ii. At what displacement, x, is the velocity, v,
greatest? What is the acceleration, a, at this
position?
Q iii. At what displacement is the acceleration
greatest? What is the velocity at this position?
Q iv. At what position is the force the greatest in
magnitude.