Resource Lesson
Period of a Pendulum
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A
simple pendulum
consists of a string, cord, or wire that allows a suspended mass to swing back and forth. The categorization of "simple" comes from the fact that all of the mass of the pendulum is concentrated in its "
bob
" - or suspended mass.
As seen in this diagram, the
length
of the pendulum is measured from the pendulum's point of suspension to the center of mass of its bob. Its
amplitude
is the string's angular displacement from its vertical or its
equilibrium position
. If a pendulum is pulled to the right side and released to swing back and forth, its path traces our a sine curve as shown below.
The time required for one complete vibration, for example, from one crest to the next crest, is called the pendulum's
period
and is measured in seconds.
The formula to calculate this quantity is
where
L
is the length of the pendulum in meters
g
is the
gravitational field strength
, or acceleration due to gravity
This quantity at sea level is 9.81 m/sec
^{2}
and can be calculated as
where
G
= 6.67 x 10
^{-11}
nt m
^{2}
/kg
^{2}
M
_{Earth}
is the mass of the earth (6.02 x 10
^{24}
kg)
R
_{Earth}
is the average radius of the earth (6.4 x 10
^{6}
meters)
Notice in the formula that the mass of a simple pendulum's bob does not affect the pendulum's period; it will however affect the
tension
in the pendulum's string.
In this
related lesson
, you will find a
derivation
of this formula for the period of a simple pendulum that will help you understand the restrictions on its use. It will also explain to you why a simple pendulum is NOT a true representation of simple harmonic motion,
SHM
. Take a few moments and use this
physlet
to investigate how the period of a pendulum is impacted by its length and its initial displacement.
The
frequency
of a pendulum represents the number of vibrations per second. This quantity is measured in hertz (hz) and is the reciprocal of the pendulum's period.
Let's practice a few problems with these formulas.
What would be the period of a pendulum located at sea level if it is 1.5 meters long?
If the pendulum's length were to be shortened to one-fourth its original value, what would be its new period?
How many complete vibrations would this shorter pendulum trace out in one minute if it were to be released with a small initial amplitude?
At sea level, how long would a pendulum be if it has a frequency of 2 hz?
The timing mechanism in a grandfather's clock is based on the principles of a simple pendulum. If your clock is gaining time, should you shorten or lengthen its pendulum?
Would a grandfather clock keep time on the moon?
A
physical pendulum
could be illustrated by swinging a meter stick about one end or a baseball bat about one end. The formula to calculate the period of a physical pendulum is
where
is the pendulum's
moment of inertia
measured in kg m
^{2}
m
is its mass in kilograms
g
is the local gravitational field strength or acceleration due to gravity
L
is the moment arm or perpendicular distance from the pivot point to object's center of mass measured in meters
Related Documents
Lab:
Labs -
A Physical Pendulum, The Parallel Axis Theorem and A Bit of Calculus
Labs -
Conical Pendulums
Labs -
Conical Pendulums
Labs -
Conservation of Energy and Vertical Circles
Labs -
Introductory Simple Pendulums
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Kepler's 1st and 2nd Laws
Labs -
Loop-the-Loop
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Moment of Inertia of a Bicycle Wheel
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Oscillating Springs
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Roller Coaster, Projectile Motion, and Energy
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Sand Springs
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Simple Pendulums: Class Data
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Simple Pendulums: LabPro Data
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Video LAB: A Gravitron
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Video LAB: Circular Motion
Labs -
Video LAB: Looping Rollercoaster
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Water Springs
Resource Lesson:
RL -
A Derivation of the Formulas for Centripetal Acceleration
RL -
Centripetal Acceleration and Angular Motion
RL -
Conservation of Energy and Springs
RL -
Derivation of Bohr's Model for the Hydrogen Spectrum
RL -
Derivation: Period of a Simple Pendulum
RL -
Energy Conservation in Simple Pendulums
RL -
Gravitational Energy Wells
RL -
Kepler's Laws
RL -
LC Circuit
RL -
Magnetic Forces on Particles (Part II)
RL -
Rotational Kinematics
RL -
SHM Equations
RL -
Simple Harmonic Motion
RL -
Springs and Blocks
RL -
Symmetries in Physics
RL -
Tension Cases: Four Special Situations
RL -
The Law of Universal Gravitation
RL -
Thin Rods: Moment of Inertia
RL -
Uniform Circular Motion: Centripetal Forces
RL -
Universal Gravitation and Satellites
RL -
Vertical Circles and Non-Uniform Circular Motion
Review:
REV -
Review: Circular Motion and Universal Gravitation
Worksheet:
APP -
Big Al
APP -
Ring Around the Collar
APP -
The Satellite
APP -
The Spring Phling
APP -
Timex
CP -
Centripetal Acceleration
CP -
Centripetal Force
CP -
Satellites: Circular and Elliptical
NT -
Circular Orbits
NT -
Pendulum
NT -
Rotating Disk
NT -
Spiral Tube
WS -
Basic Practice with Springs
WS -
Inertial Mass Lab Review Questions
WS -
Introduction to Springs
WS -
Kepler's Laws: Worksheet #1
WS -
Kepler's Laws: Worksheet #2
WS -
More Practice with SHM Equations
WS -
Pendulum Lab Review
WS -
Pendulum Lab Review
WS -
Practice: SHM Equations
WS -
Practice: Uniform Circular Motion
WS -
Practice: Vertical Circular Motion
WS -
SHM Properties
WS -
Static Springs: The Basics
WS -
Universal Gravitation and Satellites
WS -
Vertical Circular Motion #1
TB -
Centripetal Acceleration
TB -
Centripetal Force
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