The Simple Pendulum
Galileo (1564-1642) was the father of the scientific method. One of the things he
found came from watching a candelabra swinging in the Cathedral of Pisa
while attending mass. He used his pulse to measure how long it took the
candelabra to swing back and forth (the period of a pendulum). His further
studies revealed something that enabled him to make the first pendulum clock. These new
clocks were then used to see if a person’s pulse was normal.
What factors affect the period of a pendulum?
Materials: String, several different masses (500g, 250 g, 100g), stop watch ring stand and
clamp, meter stick.
Part 1: The Effect of the Mass
Procedure: Hang the one weight from a clamp attached to a ring stand using the string
provided. Measure the length of the string and record the mass (1 mass unit). Lift the mass
approximately 20 cm from the where it was at rest (the amplitude) and time how long it
takes for the pendulum to swing up and back to the original position for 5 complete swings.
Do this a couple of times until you get consistent results. Record the release height and the
time for the period of one complete swing. Repeat with two weights then three weights and
record your results.
Part 2: The Effect of the Amplitude
Procedure: Using a constant mass (one weight) increase the amplitude to 40 cm above the
rest position. Do this a couple of times until you get consistent results and record your
results for 5 complete swings. Repeat with an amplitude of about 60 cm. Be sure that the
mass does not strike anything while it swings and that it swings straight and not wobble.
Part 3: The Effect of the Length of String
Procedure: Using the same mass (one weight) and the same amplitude (20 cm) shorten the
string and record the new length. Swing the pendulum with the same amplitude each time
and record the period of 5 oscillations a couple of times until you get consistent results.
Repeat with an even shorter string. Compare to the period of the string with the same mass
and amplitude with a full-length string.
Question: From your data, what factor affects the period of a swinging pendulum?
Part 4: The Acceleration due to Gravity
The equation that relates the period and length of a pendulum is given by the formula:
T = 2 π √L/g
Rearrange this formula to solve for g and substitute in the period and length for your
pendulum data.
The Pendulum
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PART II: POTENTIAL & KINETIC ENERGY IN A PENDULUM
If a pendulum is pulled to some angle from the
vertical but not released, potential energy exists in
the system. When the pendulum is released, the
potential energy is converted into kinetic energy as
the pendulum bob descends under the influence of
gravity. The faster the pendulum bob moves, the
greater its kinetic energy. The higher the
pendulum bob, the greater its potential energy.
This change from potential to kinetic energy is
consistent with the principle of conservation of
mechanical energy which states that the total
energy of a system, kinetic plus potential, remains
constant while the system is in motion.
Maximum
Maximum
GPE
GPE
Maximum KE
When you pull the pendulum to the side, you increase the gravitational potential energy of
the pendulum by an amount equal to the change in height times the mass times the
acceleration of gravity. So we can write GPE=m g h, where GPE is the change in potential
energy, m is the mass in kilograms, h is the vertical distance that the pendulum has been
raised, and g is 9.80 m/s² as before.
Kinetic energy of motion is given by the formula K E= ½ m v², where m is mass in kilograms,
and v is the velocity of the pendulum in m/s. If the energy is conserved, all of the potential
energy at the top of the swing should be converted to kinetic energy at the bottom of the
swing where the velocity is greatest. Let's test this.
PROCEDURE
In this portion of the experiment, you will test whether energy is conserved in a pendulum
by using a photogate timer that measures the time it takes the falling bob to pass through a
narrow beam of light. From this the speed of the falling bob can then be calculated.
Comparing the kinetic energy at the bottom of the swing with the amount of potential
energy at the release point will test the conservation of energy of the pendulum.
Make the following measurements for your pendulum and record the data in the table
below:
Mass of bob:
g
kg
Diameter of bob:
cm
m
Height of bob at rest above table:
cm
m
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You will collect the time it takes for the bob to pass through the photogate for 3 trials at two
different release heights. Pull back the pendulum and measure the height of the bob above
the table using a ruler. Try to keep the height of the bob the same for each of the three
trials. Reset the timer between trials.
Release Height
Time: Trial 1
Time: Trial 2
Time: Trial 3
Average
40.0 cm
20.0 cm
How much higher (vertically) is the pendulum at each release height than it was
when it was hanging at rest? Convert this distance to meters and calculate the
gravitational potential energy, GPE, of the bob.
Gravitational Potential Energy at Release Point
40.0 cm Release Height
20.0 cm Release Height
Calculate the velocity of bob at the bottom of the swing:
diameter of bob (m)
= velocity of bob (m/s)
average time (sec)
40.0 cm Release Height
Velocity at the Bottom
20.0 cm Release Height
Calculate the kinetic energy of the bob at the bottom of the swing.
40.0 cm Release Height
Kinetic Energy at Bottom
20.0 cm Release Height
Compare the values for the gravitational potential energy and kinetic energy of the
pendulum. Was energy conserved? That is, were they equal? If not, how might you
account for the difference in energies?
The Pendulum
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