PYL 106: Pendulum
The Pendulum
A mass hanging from a string or attached to a rigid rod, i.e. a pendulum, is another
example of a system which exhibits periodic motion. (In such a case, the mass is known
as a bob.) The equilibrium position is when the string or rod hangs vertically. If the
pendulum is not in this position, there is a restoring force. One of the quantities
studied for a pendulum is the angle between the string and the vertical. Remember the
angles we are talking about will be measured in radians.
Recall that the period T is the time taken before the motion starts repeating
itself. We want to study the effect on the period of varying
 the mass of the bob
 the length of the string
 the amplitude, i.e. the maximum angle between the string and
the vertical
(Note that we will use a mass on the end of a string to study the effect of mass and length
variations, and we will use a rigidrod pendulum to study the amplitude variations.) You
should make sure that when varying one parameter, the others are held fixed to whatever
extent possible.
The forces acting on the bob of a pendulum are its weight and the tension of the string. It is
useful to analyze the pendulum in the radial/tangential
coordinate system. The tension lies
completely in the radial direction and the weight must be broken into components.
The net radial force leads to radial acceleration, which is a
centripetal acceleration.
where the radius of the circular motion is
l, the length of the pendulum.
(Recall the formula for centripetal acceleration is v^{2}/r.) The tension and the centripetal
acceleration are both directed toward the center of the circle.
The net tangential force leads to a tangential acceleration.
where the tangential acceleration is l
α and
α is the
angular acceleration, d^{2}θ/dt^{2}. The negative sign
indicates that the net force is a restoring force, i.e., that the tangential
force is in the opposite
direction of the displacement from equilibrium θ.
Another approach to the pendulum is conservation of energy. At an
arbitrary angle, the energy is a combination of kinetic energy and
gravitational potential energy. At the highest point of the swing,
the energy is entirely potential; at the lowest point of the swing, the
energy is entirely kinetic (if we take the lowest point to have zero
gravitational potential energy).
Substituting this expression for v_{max} which occurs at the equilibrium
position into the expression for the tension at the equilibrium position θ=0 yields the following expression for the maximum tension:
which will be tested toward the end of the experiment.
Experiment

Plug a photogate into Digital Channel 1 of the DataStudio/Pasco interface.

Start Data Studio, click on the digital plug into the image of interface in
Data Studio, and choose Photogate & Pendulum from the menu.

Using a ringstand, clamps, a mass, some string, and so on, set up a pendulum
as shown below.

Make sure that as the bob is moving, it passes through the center
of the Ushaped part of the photogate (or more importantly the little lights in
the photogate).

Initiate the pendulum's motion, record data for a few periods.

Drag a table into Digital Channel 1 and choose Period from the menu.
 Vary the bob mass.

Tie bobs of different masses to the end of your string. Use
the cylinders of the same size but different mass.

To the best of your ability, make sure that the distance between
the length of the string from the crossbar to the center of
mass of the bob remains fixed.

Make sure you release the bob from the same initial angle, you may
want to use a protractor.

Record the masses and the periods for four different bobs.
 Vary the pendulum length.

Vary the length of the pendulum. To ensure a substantial
range of variation, your smallest length should be less than 30 centimeters and your longest length should
be close to 1 meter.

Keep the mass and amplitude fixed.

Record the lengths and periods for five different lengths.
 Vary the amplitude.

Replace the massonastring pendulum with a rigidrod
pendulum, which is shown below. There is a protractor on the pendulum and the
angle on that protractor should be fixed at 90°.

Release the pendulum from different initial angles:
5°,
15°,
25°,
35°,
45°, and
55°. (These angles will be measured
by a separate protractor and not the one that comes with the pendulum.) If 5° is too small to
obtain a good measurement, then switch to 10°.
 Record the corresponding periods.
Amplitude Variation
Amplitude (°) 
Period ( ) 
5  
15  
25  
35  
45  
55  
Maximum tension

Close your Data Stduio file and start a new one.

Replace the photogate with a force sensor. Force sensors
are plugged into the Analog channel.

Click on the Sampling Options button. Set the sampling
frequency to 20 Hz.

Attach the force sensor to a pole on a ringstand as shown
below.

Tare the force sensor while nothing is hanging from it.
(There's a Tare button on the side of the Force sensor.)

Tie the pendulum string to the force sensor as shown below.

Attach one of the hexagonal shaped masses to serve as the
pendulum bob.

With the mass in its equilibrium position, take a force sensor reading.
This reading will be the weight mg as measured by the force sensor.

Release the mass from an angle of 20°,
record a few periods of data.
Analysis.

Plot the period versus mass. What kind of dependence if
any do you see?

Plot the period versus length. Fit it to a power law. What
power do you find? Does it agree with the theoretical prediction found
in your book?

Plot the period versus amplitude. Compare your results with
the theoretical prediction found in your book. Do you observe any
systematic deviations from the predicted value for large angles? The
derivation of this formula assumes that the angle is small, so that
sin(θ) ≈ θ,

Make a table of θ (in radians!)
and sin(θ) for the amplitudes
used for the above measurements.
θ (in degrees) 
θ (in radians) 
sin(θ) 
¼ sin^{2} (θ/2) 
5    
15    
25    
35    
45    
55    
According to Mathematical Methods for Physicists (G. Arfken, p. 322
in the third edition)
where θ_{M} is the amplitude.
According to this equation, does the period increase or decrease when
the amplitude is increased? Does this agree with your results?

Using the force sensor data, plot Tension vs time.
Extract the tension with the maximum amplitude, which will be at
the bottom of the graph. It is approximately 5.4 N in the
example shown below.
Compare T_{max}/mg to the theoretical prediction of
3  2 cos(θ).
Weight from Force Sensor when still ( ) 
T_{max} from Force Sensor when swinging
( ) 
T_{max}/mg 
3  2 cos(θ) 
Percent diff. 
    

A "seconds" pendulum moves through its equilibrium position
(the vertical) every second. That means that its period is
2.00 s. If you had to build a seconds pendulum, what would you
use for its length, bob mass and amplitude?