Transcript Simple Harmonic Motion

Simple Harmonic Motion
AP Physics C
Simple Harmonic Motion
What is it?
 Any periodic motion that can be
modeled with a sin or cosine wave
function.
 Harmonic oscillators include:
 Simple pendulum – a mass swinging on
a string or rod
 Mass on a spring that has been offset
from its rest position and then released
Reminders: Hooke’s Law and
Conservation of Energy
F  kx
U el 
1
2
kx
Hooke’s Law
2
Ki  U i  K f  U f
Elastic Potential Energy
Conservation of Energy
Waves are oscillations too!
Remember Wave Characteristics
 Amplitude
 the maximum displacement of the medium measured
from the rest position.
 Wavelength
 The distance between corresponding points on
consecutive waves.
 Frequency
 the number of complete cycles (waves) that pass a
given point in the medium in 1 second.
 Period
 the time it takes for one complete cycle to pass a given
point in the medium, or the time that passes before the
motion repeats itself
 Wave Speed
 velocity the wave travels through a medium
Simple Harmonic Motion (SHM)
velocity & acceleration
Note: k= spring constant
K=kinetic energy,
Position (for reference only)
T=period
a=max
A
v=0
v=max
B
a=0
a=max
C
v=0
a=0
a=max
E
At t=T/4: X = 0 (no disp), F=0 (at
equilibrium), a=0 (no force), v=max,
Uel=0 (no stretch), K=1/2 mV2 .
At t=T/2: X = A (on other side of x=0),
F=kA (toward x=0), a=kA/m (max),
v=0, Uel=1/2 kA2, K=0.
v=max
D
At t=0: X = A (max disp), F=kA
(toward x=0), a=kA/m (max), v=0,
Uel=1/2 kA2, K=0.
v=0
At t=3T/4: X = 0 (no disp), F=0 (at
eq.), a=0 (no force), v=max (opp. dir.),
Uel=0 (no stretch), K=1/2 mv2 .
At t=T: Back to original position so
same as t=0, X = A (max disp), F=kA
(toward x=0), a=kA/m (max), v=0,
Uel=1/2 kA2, K=0.
More velocity & acceleration in SHM
What happens between x=0 and x=A?
Note: F & a are always
directed toward x=0 (eq)
a=max
v=0
a
v
v
a=max
v=0
x=A
x=A
v=max
a
(t=0 to t=T/2)
Between t=0 and t=T/4, the
mass is moving toward the
equilibrium position (from x=A
to x=0) with a decreasing
force. a , v and a in same dir
so v . Uel , and K .
Between t=T/4 and t=T/2, the
mass is moving away from the
equilibrium position (from x=0
to x=A) with an increasing
force. a , v and a in opp dir, so
v . Uel , and K
More velocity & acceleration in SHM
What happens between x=0 and x=A?
Note: F & a are always
directed toward x=0 (eq)
a=max
Between t=T/2 and t=3T/4, the
mass is moving toward the
equilibrium position (from x=A
to x=0) with a decreasing
force. a , v and a in same dir
so v . Uel , and K .
a
v
x=A
a=0
x=A
(t=T/2 to t=T)
a
a=max
v
Between t=3T/4 and t=T, the
mass is moving away from the
equilibrium position (from x=0
to x=A) with an increasing
force. a , v and a in opp dir, so
v . Uel , and K .
Angular Frequency
For SHM we define a quantity called angular
frequency, ω (which is actually angular velocity),
measured in radians per second. We use this
because when modeling the SHM using a cosine
function we need to be able to express frequency
in terms of radians.
if frequency, f is in hertz (
  f
rev
s

2 rads
rev
  2 f
rev
s
)
Modeling SHM with a cosine wave.
When we start an oscillation, such as a mass on a spring, we either
stretch or compress a the spring a certain distance which then
becomes the Amplitude of the oscillation.
x(t )  A cos(t   )
Where A=Amplitude, ω=angular
frequency, t=time, and φ = phase shift
Note: when t=0, x=A (max displacement)
Modeling velocity for SHM
d
v(t )  x(t )
dt
v(t )   A sin(t   )
Notice that at t=0, v=0 and this
corresponds to the maximum
displacement (x=A).
Also…the maximum value occurs
at sin(∏/2)=1, so vmax=-ωA
Modeling acceleration
and finding maximum acceleration
d
a(t )  x(t )
dt
a(t )   2 A cos(t   )
Notice that at t=0, a=max in the
opposite direction as the stretch
and this corresponds to the
maximum displacement (x=A).
Also…the maximum value occurs at
cos(0)=1, so amax=-ω2A
Calculating acceleration
for a mass on a spring
F   kx  ma
k
so... a   x
m
This means that acceleration
is a function of position.
Or… more stretch means more
acceleration.
We will express acceleration in a general for in terms of
angular frequency, ω, and position, x, as shown.
a   2 x
k
k
so if   , then  
m
m
2
Note: at x=A, a=max = -ω2A
Simple Pendulum
A simple pendulum is an object of mass,
m, swinging in a plane of motion,
suspended by a massless string or rod.
θ
l
In other words, it is a point mass moving
in a circular path.
l
If θ < 10°, then we can assume a small
angle approximation, sin θ ≈ θ, the long
formula for period, which includes an
infinite sine series, reduces to…
T  2
Note:

g
g