Resource Lesson
Derivation: Period of a Simple Pendulum
Printer Friendly Version
Simple pendulums are sometimes used as an example of simple harmonic motion,
SHM
, since their motion is periodic. They also fit the criteria that the bob's velocity is maximum as it passes through equilibrium and its acceleration is minimal while at each endpoint. But a deeper understanding of the behavior of the bob will show us that pendulums do not truly fit the SHM model.
To begin our analysis, we will start with a study of the properties of force and acceleration in a simple pendulum by examining a freebody diagram of a pendulum bob.
As the pendulum swings, it is accelerating both centripetally, towards the point of suspension and tangentially, towards its equilibrium position. It is its linear, tangential acceleration that connects a pendulum with simple harmonic motion.
The weight component, mg sinθ, is accelerating the mass towards equilibrium along the arc of the circle. This component is called the
restoring force of the pendulum
.
F
_{restoring}
= ma
_{tangential}
mg sinθ = ma
_{tangential}
To strictly qualify as SHM this restoring force should be directly proportional to the bob’s linear displacement from equilibrium along the length of the chord.
Geometrically, the arc length,
s
, is directly proportional to the magnitude of the central angle,
θ
, according to the formula s = rθ. In our diagram the radius of the circle, r, is equal to L, the length of the pendulum. Thus, s = Lθ, where θ must be measured in radians. Substituting into the equation for SHM, we get
F
_{restoring}
= - ks
mg sinθ = - k(Lθ)
Solving for the "spring constant" or
k
for a pendulum yields
mg sinθ = k(Lθ)
k = mg sinθ / Lθ
When an angle is expressed in radians, mathematicians generally represent the angle with the variable
x
instead of
θ
. Note that the value of
sin x
approximates the value of
x
for small angles; that is,
Or, equivalently, for
x
equal to small values you can see from this power series that the value of
sin x
would approach that of
x
.
Using this relationship allows us to reduce our expression for the pendulum's "spring constant" to
k = mg / L
Substituting this value for
k
into the SHM equation for the period of an oscillating system results in
Before leaving this lesson, let's examine our formula's error for a pendulum released at an angle of 10º.
10º = 0.1745 radians
sin(0.1745) = 0.1736
Our calculated value for the period will be 0.26% too high. An error that most would readily accept in lieu of using a more complicated formula.
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.
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
Labs -
Kepler's 1st and 2nd Laws
Labs -
Loop-the-Loop
Labs -
Moment of Inertia of a Bicycle Wheel
Labs -
Oscillating Springs
Labs -
Roller Coaster, Projectile Motion, and Energy
Labs -
Sand Springs
Labs -
Simple Pendulums: Class Data
Labs -
Simple Pendulums: LabPro Data
Labs -
Video LAB: A Gravitron
Labs -
Video LAB: Circular Motion
Labs -
Video LAB: Looping Rollercoaster
Labs -
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 -
Energy Conservation in Simple Pendulums
RL -
Kepler's Laws
RL -
LC Circuit
RL -
Magnetic Forces on Particles (Part II)
RL -
Period of a Pendulum
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
PhysicsLAB
Copyright © 1997-2015
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton