Simple Harmonic Motion - schools.misd.orgschools.misd.org/upload/template/6302/Mont SHM with springs...
Transcript of Simple Harmonic Motion - schools.misd.orgschools.misd.org/upload/template/6302/Mont SHM with springs...
Simple Harmonic Motion
Definitions
• Periodic Motion – When a vibration or
oscillation repeats itself over the same
path
• Simple Harmonic Motion – A specific form
of periodic motion in which the restoring
force is proportional to distance from the
equilibrium position.
Objects that exhibit SHM
• Spring Systems
• Pendulums
• Circular Motion
• Waves
– Sound, Light, Pressure
Definitions
• Period – Time required for one complete
cycle (Seconds).
• Frequency - Number of complete cycles
in a period of time (Hz).
• Amplitude – Displacement from the
equilibrium position. It is a measure of the
energy of an oscillator (Different Units).
Definitions
• Equilibrium Position - The center of
motion; the place at which no forces act.
• Displacement - The distance between the
center (equilibrium position) and location
of the wave at any time.
• Restoring Force – The force exerted on
the medium to bring it back to the
equilibrium position.
Some Visual Aids!
Graphical Representation
• Sine curves describe SHM very well!
y = Asin(wt)
Relating Period and Frequency
• The period and the frequency always have
the same definition, regardless of the topic
being discussed.
• Remember, these two values are inverses
when in SI units:
f = 1/T T = 1/f
Example
Terry jumps up and down on a trampoline
with a frequency of 1.5 Hz. What is the
period of Terry’s jumping?
f = 1.5 Hz
T = 1/f = 1/(1.5) Hz
T = .67 s
Springs
• The object to the left is a
common spring system.
• It has a mass of some
kind attached to a spring.
• This spring is stretched
and released. This
causes the entire system
to oscillate.
Springs • The spring supplies the
restoring force on the
mass.
• As the mass gets further
away from the equilibrium
position, the force upon it
gets greater.
• There is no force on the
mass at the equilibrium
position.
Springs
• So the equation for force of a spring is as
follows:
F = kx
(Hooke’s Law)
F – the force supplied by the spring
k – the spring constant (depends on how the
spring is made)
x – displacement of the spring from its
equilibrium position
Example 2
I have a slinky with a spring constant of
100 N/m. If I stretch the slinky 5 meters
from its equilibrium position, with what
force will the spring pull on my hand?
F = kx
F = (100 N/m)(5m) = 500 N
F = 500 N
Period
• If we stretch a spring with a mass and
release it, it will oscillate.
This is SHM!
What is the period of this
Motion?
Period
• The period of a spring system is given by
the equation below:
T =
T – the period of motion
m – Mass of the body attached
k – spring contant
km /2
Period
• There are some important things to notice
about this equation:
1. The larger the mass attached to the
spring, the longer the period.
2. The stronger the spring, the shorter the
period.
3. Remember that period is always in
seconds!
Example
• What is the mass of my car if the shocks
have a spring constant of 6000 N/m and it
oscillates with a period of 2 seconds when
I hit a bump in the road?
T =
(T/2π)2 = m/k
(6000 N/m)(2 s/2π)2 = m
607.9 kg = m
km /2
Pendulums
The object on the left is a
pendulum.
It is usually a mass
hanging on the end of a
string.
It could also be a mass on
the end of a pipe or
other variation.
Pendulums
Gravity supplies the
restoring force to create
Simple Harmonic
Motion.
Note that the higher the
pendulum goes, the
more gravity acts to
bring the pendulum back
to its equilibrium
position.
Restoring Force
The restoring force for
a pendulum is the
component of gravity
that is tangent to its
circular motion.
In most cases:
F = mgsin(Θ)
Example
• What restoring force does gravity supply to
a 0.5 kg pendulum that is at 30 degrees?
F = mgsin(Θ)
F = (.5 kg)(10 m/s2)sin(30)
F = 2.5 N
• What is the restoring force at 90 degrees?
F = mgsin(Θ)
F = (.5 kg)(10 m/s2)sin(90)
F = 5 N
Period
• The period of a pendulum is given by the
following formula:
T =
L – Length of the Pendulum
g – Acceleration due to Gravity
T – Period
gL /2
Period
• There are some important things to notice
here:
1. The period does not depend on the
mass of the pendulum bob.
2. The length and gravity are the only
values that affect the period.
3. As the length of a pendulum becomes
longer, the period becomes longer.
Example
• I find an old grandfather clock with a period
of 1.0 s per swing. If the grandfather clock
is located in Montgomery (g = 10 m/s2),
how long is the pendulum?
T =
(T/(2π))2 = L/g
g(T/(2π))2 = L
(10 m/s2)(1s/(2π))2 = L
0.25m
gL /2