Chapter 12 –Vibrations and Waves
12-1 Simple Harmonic Motion
page 438-451
Hooke’s Law
Periodic motion – the object has a repeated motion that follows the same path, the object
swings to and fro.
Examples: a pendulum on a clock; play ground swing; a diving board
The simplest harmonic motion is the back and forth periodic motion of a mass attached to a
spring. When the spring is stretch or compressed then released, it vibrates back and forth
around its unstretched position.
The Vibrating Spring
Vibrating spring facts:
1. Equilibrium Position
At equilibrium position where X = 0 the velocity is maximum and X = 0 is the position of
rest of the spring. When the spring is either compressed or stretched then released, the
spring will “spring” back towards the equilibrium position. As the energized spring
approaches its equilibrium position (rest) due to maximum velocity at that point, it will
over shoot the rest position.
2. Maximum Displacement
At maximum displacement (compression / maximum stretch at some ± distance from x=0) the
spring force and acceleration are at maximum. When the compressed spring is released, it
“springs” back toward its equilibrium position but it will overshoot that position. The distance
of overshoot will be the maximum displacement.
In an ideal frictionless mass-spring system the system would oscillate (vibrate back-and-forth)
indefinitely. The friction will retard the oscillating in the system and the friction will
eventually bring the system to rest. This effect is called damping. The damping effect is
minimal over a short period of time.
3. Restoring force is proportional to displacement
The force, the restoring force, that is exerted by the spring always pushes or pulls the
mass towards the original equilibrium position. The restoring force is always directly
proportional to the mass. Any periodic motion that results from the restoring force that’s
proportional to displacement is described as simple harmonic motion.
Simple harmonic motion – vibration around an equilibrium position in which a restoring
force is proportional to the displacement from equilibrium.
1
Hooke’s Law – mass-spring systems obey a simple relationship between force and
displacement. For small displacements:
Spring Force = -(spring constant x displacement)
or
Felastic = -kx
Felastic – Spring force (Newton N)
k – spring constant (SI units of k are N/m)
x – displacement (meters)
Notes for Hooke’s Law formula:
1. Spring constant (k) is always a positive number.
2. A greater constant means a stiffer spring and requires more force to compress or stretch the spring.
3. The negative sign indicates the direction of the spring force is opposite the direction of the mass’s
displacement from equilibrium.
4. Stretched or compressed spring has elastic potential energy
A stretch or compressed spring stores elastic potential energy. When the spring is
stretched, it stores elastic potential energy and the kinetic energy is zero because the
system is at rest. When the spring is released the springs’ elastic potential energy is
converted into kinetic energy. Due to conservation of energy, the kinetic energy is equal to
the stored elastic potential of the system.
http://www.ux1.eiu.edu/~cfadd/1150/15Period/SHM.html
2
The Simple Pendulum
A pendulum consists of a mass (bob) that is attached to a fixed string that is free to
swing back-and-forth under the influence of gravity. Examples include a clock
pendulum, trapeze artists, and a swing. (Have you ever read Poe’s famous predicament short
story?) The equilibrium position of a pendulum is when the bob is hanging straight
down.
When working with a pendulum assume the mass of the bob is concentrated at the end
of the string and disregard the mass of the string, friction, and air resistance during the
analysis of the pendulum.
Swinging Pendulum Facts:
1. Restoration force of a pendulum
Establish a coordinate system, x-axis will always be tangent to the direction of motion
and y-axis will always be perpendicular to the direction of the motion. Therefore the
coordinate system will always change due to the changing position of the bob.
Forces on the bob: string (always along the y-axis), weight (gravity).
The bob’s weight can be resolved into two components. Since the forces exerted b the
strung and the y component of the bob’s weight are perpendicular to the bob’s motion, the
x component of the bob’s weight is the net force on the bob in the direction of the bob’s
motion. Therefore the restoring force of the pendulum is the x component of the bob’s
weight because the weight always pushes of pulls the bob toward its equilibrium position.
2. Small angles of pendulum motion
The pendulum’s motion is simple harmonic motion. The restoring force of a simple
pendulum is not constant because the restoring force varies with the bob’s distance from
the equilibrium position. The magnitude of the restoring force decrease each time the bob
moves toward the equilibrium position.
When the angle is relatively small (<15°) the restoring force is proportional to the
displacement and as the pendulum swings back-and-forth, it displays simple harmonic
motion.
3. Gravitational Potential influence on the pendulum’s motion
As the string swings past its’ equilibrium position, the kinetic energy decreases while the
gravitational potential energy increase. At the instant the pendulum is at the peak of travel
away from (maximum displacement) the equilibrium position, its KE is zero and the PE is
maximum. When the pendulum is at its equilibrium position, its PE is minimum and the KE
is maximum. The Pendulums’ energy is dependent on the influence of gravity.
3
12 –2 Measuring Simple Harmonic Motion
page 446
Amplitude, Period, and Frequency
As the pendulum swings back and forth, maximum distance from equilibrium is the
amplitude and the amount of time it takes the pendulum to swing back and forth one
time is the period. The number of periods per unit of time is the frequency.
Amplitude – the maximum displacement from equilibrium; also release height or angle
Period (T) – the time it takes to complete one back-and-forth cycle of motion
Frequency (f)– the number of cycles or vibrations per unit of time (SI unit is Hertz – Hz – the
number of periods per second)
Bob – the concentrated mass at the end of the string
Length – the length of the string (m) measured from the pivot point to the center of gravity
of the mass
The basic formulas that describe the relationship between frequency and period are:
f=
1
T
or
T=
1
f
Note that period is time per cycle and frequency is cycles per unit of time. These two
variables are inversely related.
Sample problems:
a) You count 12 ocean waves rolling past the same point in one minute. What is the
frequency?
b) What is the frequency if the period is 0.01666?
1. The period of a simple pendulum depends on pendulum length and free-fall
acceleration.
Both the simple pendulum and mass-spring system will vibrate with simple harmonic motion
but each requires a different equation because the period and frequency depend on
different physical factors.
Factors that affect the period of a simple pendulum:
a. Change the length of a pendulum.
b. Change in free-fall acceleration.
c. Mass does not affect the period.
4
Formula for period of a simple pendulum in simple harmonic motion:
T = 2π
L
g
T = period
L = pendulum length
g = free-fall acceleration
Two pendulums with different lengths but same amplitude:
Shorter pendulum will have smaller arc length.
Shorter pendulum will have less displacement.
Shorter pendulum will have shorter period.
Both pendulums will have the same free-fall acceleration.
Sample problems:
a) It takes a pendulum 2 seconds to complete one cycle of motion. What is the length of the
pendulum?
b) If a pendulum has a length of 150 cm, what is the period?
c) If a pendulum has a frequency of 5 Hz, what is the period? What is the string length?
2. Period of a mass-spring system depends on mass and spring constant.
Review Hooke’s law. The restoring force acting on the mass is determined by the
displacement of the mass and the by the spring constant. The magnitude of the mass does
not affect the restoring force.
What is affected by the magnitude of the mass is the moment of inertia and the
acceleration. The larger mass will have a smaller acceleration and the larger mass will have
a greater period or it takes a longer time to complete one cycle of vibration.
Note that a stiffer spring has a greater spring constant. A greater spring constant means
that more force is required to compress or stretch the spring. A greater spring constant also
indicates an increased acceleration and a less time for a completed single cycle.
Formula for period of a mass-spring system in simple harmonic motion:
T = 2π
m
k
T = period
m = mass in kg
k = spring constant
5