Download SHM notes - Sign in to St. Francis Xavier Catholic School System

Simple Harmonic Motion
Any regular vibrations or oscillations
that repeat the same movement on
either side of the equilibrium position
and are a result of a restoring force
Restoring force: a force that tries
to return an object to equilibrium
(center resting position).
Results in back and forth motion
over the same path
Amplitude
Maximum displacement from equilibrium
• Pendulum: measured by the angle between the pendulum’s
equilibrium position and its maximum displacement.
• Mass-spring system: maximum amount the spring is
stretched or compressed from its equilibrium position.
• Units: radian (rad) or degrees and the meter (m).
Period
Frequency
• Time it takes to complete
one cycle
• Units: Seconds
• Number of cycles or
vibrations per unit of time
• Units: hertz (Hz)
Period & Frequency are inversely related
1
1
f  or T 
T
f
Period vs. Amplitude
Period vs. Mass
Period vs. Length
Period squared vs. Length
Period of a Simple Pendulum in
SHM
• The period of a simple pendulum depends on
the length and on the free-fall acceleration.
length
period  2
free-fall acceleration
L
T  2
ag
2


4

2
T 

 g 
The period does not depend on the mass of the bob or on the amplitude
(for small angles).
Kinetic Energy
• energy of an object due to the object’s
motion
• depends on speed and mass.
1
KE  mv 2
2
1
2
kinetic energy =  mass   speed
2
Potential Energy
• Stored energy associated with an object
because of the position, shape, or condition of
the object.
• Gravitational potential energy is the energy an
object has because of its position in a
gravitational field.
• GPE depends on height from a zero level and
the mass of the object.
PEg = mgh
gravitational PE = mass  free-fall acceleration  height
Potential Energy, continued
• Elastic potential energy is the energy
available for use when a deformed elastic
object returns to its original configuration.
PEelastic
elastic PE =
1
1 2
 kx
2
 spring constant  (distance compressed or stretched)
2
•
The symbol k is called the spring constant, a
parameter that measures the spring’s resistance to
being compressed or stretched.
2
Mechanical Energy
• The sum of kinetic energy and all forms of
potential energy associated with an object
or group of objects
ME = KE + ∑PE
• Is often conserved
MEi = MEf
initial mechanical energy = final mechanical
energy (in the absence of friction)
Energy in pendulums & springs
SHM in springs
• The direction of
the force acting
on the mass
(Felastic) is
opposite the
direction of the
mass’s
displacement
from equilibrium
(x = 0).
SHM in hanging spring
SHM in springs
At equilibrium:
•
•
•
•
Spring force?
Speed?
Kinetic Energy?
Elastic Potential Energy?
At maximum displacement:
• Spring force?
• Speed?
• Kinetic Energy?
• Elastic Potential Energy?
Hooke’s Law
• The spring force, or restoring force, is
directly proportional to the displacement
of the mass.
• This relationship is known as Hooke’s Law:
Felastic = kx
spring force = spring constant  displacement
Practice Questions
1. If a mass of 0.55 kg attached to a vertical spring stretches the spring
2.0 cm from its original equilibrium position, what is the spring
constant?
2. Suppose the spring from above is replaced with a spring that stretches
36 cm from its equilibrium position.
• What is the spring constant?
• Is this spring stiffer or less stiff?
3. How much force is required to pull a spring 3.0 cm from its equilibrium
position if the spring constant is 2700 N/m?
Period of a Mass-Spring System
• The period of an ideal mass-spring
system depends on the mass and on the
spring constant.
m
T  2
k
mass
period  2
spring constant
The period does not depend on the amplitude.
Practice: Mass-Spring System
1) A 125 N object vibrates with a period of 3.5 s when hanging from a
spring. What is the spring constant of the spring?
2) A spring of 30.0 N/m is attached to different masses, and the system
is set in motion. Find the period and frequency of vibration for
masses of 2.3 kg.
3) A child’s toy that is made to shoot ping pong balls consists of a tube,
a spring (k = 18 N/m) and a catch for the spring that can be released
to shoot the balls. When a ball is loaded into the tube, it compresses
the spring 9.5 cm. If you shoot a ping pong ball straight out of this
toy, what is its maximum speed?
Section Review
•
Which of these periodic motions are simple harmonic?
– a child swinging on a playground swing (θ = 45)
– a CD rotating in a player
– an oscillating clock pendulum (θ = 10)
•
A pinball machine uses a spring that is compressed 4.0 cm to launch a
ball. If the spring is 13 N/m, what is the force on the ball?
•
How does the restoring force acting on a pendulum bob change as the bob
swings toward the equilibrium position? How do the bob’s acceleration
(along the direction of motion) and velocity change?
Section Review
• A child swings on a playground swing with a 2.5 m long
chain.
– What is the period and frequency of the child in motion
• A 0.75 kg mass attached to a vertical spring stretches the
spring 0.30 m.
– What is the spring constant?