Simple Harmonic Oscillator and SHM
A Simple Harmonic Oscillator is a
system in which the restorative
force is proportional to the
displacement according to Hooke’s
Law:
F(x) = - kx
k is the force constant
x (even though this appears vertical) is the
displacement of object from its equilibrium
position--- F is + when x is neg. and vice versa!
Because the
force will
vary with x,
so does the
acceleration
and velocity!
Frequency (f or γ) : number of cycles per second
[Hz]
Period (T): time to complete one full cycle [s]
f = 1/ T
+x
t
-x
x = xmcos(t + ø)
  the angular frequency of the motion (rad/s)
Angular frequency differs from the frequency by
a factor of 2π :
 = 2π 
xm = maximum displacement (amplitude [A])
ø = phase constant (position of the object when
at t = 0 s – phase shift of the cosine curve)
Phase and phase constant:
x = xmcos(t + ø)
Phase: the angle over which the function repeats
itself-- (t + ø)
Phase constant (ø) is the shift of the function if
x ≠ xm at t = 0 s
ø=0
ø = π/2
The graph of an object moving in SHM is shown.
The object has a maximum displacement of 6.3 m
and is displaced 2.9 m at t = 0 s. What must be the
phase constant of this motion?
A = 6.3 m
x(t) = Acos(ωt + ø)
2.9 = 6.3cos(ω(0) + ø)
ø = cos-1 (.46)
ø = 1.1 rad or 5.2 rad
Because the graph is
increasing after t = 0
s the correct value is
5.2 rad
Displacement:
Velocity:
x(t) = Acos(t + ø)
v(t) = -Aωcos(t + ø)
Acceleration:
vmax = Aω
a(t) = -Aω2cos(t + ø) = -ω2x
amax = Aω2
F = ma = -mω2x
- mω2x = - kx
ω=
k
m
T = 2π
T = 2π m

k
A body oscillates with simple harmonic motion such
that x = (6.12 m)cos[(8.38 rad/s)t + 1.92 rad]
For t = 1.90 s, find A) the displacement B) the velocity
C) the acceleration D) the frequency and E) the period.
The scale of a spring balance is marked from 0 to 25.0
N over 14.0 cm. A package suspended from the
balance oscillates with a frequency of 2.00 Hz. What
is the weight of the package?
A 2.14 kg object hangs from a spring. A 325 g body is
attached to the object and the spring stretches an
additional 1.80 cm. The 325 g mass is removed and
the spring is set into oscillation. Find the period of
oscillation.
A block is on a horizontal surface (a shake table)
that is moving horizontally with a simple
harmonic motion of frequency 2.35 Hz. The
coefficient of static friction between the block and
the table surface is .630. How great can the
amplitude be if the block does not slip along the
surface?
If the block is not slipping then it is
experiencing SHM and Fmax  or = to fs
Fmax = mamax
a = 2xmax
answer = 2.83 cm
In harmonic motion (including SHM), ignoring
any dissipative forces, the mechanical energy is
conserved:
E=K+U
In SHM, x = xmcos(t + ø), so Pot. Energy is:
U = .5kx2 = .5kxm2 cos2(t + ø)
K = .5mv2 = .5m 2xm2 sin2 (t + ø)
At the point of max
displacement, the velocity of
E = .5kxm2
the object will be 0 and
therefore:
An oscillating block-spring system has mechanical
energy of 1.18 J, an amplitude of 9.84 cm and a
max speed of 1.22 m/s. Find A) the force constant
of the spring B) the mass of the block and C) the
frequency of oscillation.
A 5.13 kg object moves on a horizontal frictionless
surface under the influence of a spring with a force
constant of 9.88 N/cm. The object is displaced 53.5
cm and given an initial velocity of 11.2 m/s back
toward the equilibrium position. Find A) the
frequency of motion B) the initial potential energy
of the system C) initial kinetic energy and D)
amplitude of the motion.
Applications of SHM
1) The Torsional Oscillator: A disk suspended
by a wire and is twisted:
When twisted, the
tension provides a
restoring torque:
τ = -kθ
T = 2π I
k
2) The Simple Pendulum-- all of the mass is
located in the bob:
When the
displacement angle
is small (< 15˚) :
T = 2π L
g
3) The Physical Pendulum-- an oscillating body:
T = 2π I
mgh
h is dist.
from cm to
PP
A circular hoop with a mass of 2.16 kg and a
radius of 65.3 cm is suspended from a horizontal
nail. What is the frequency of oscillation for small
displacements from equilibrium?
A physical pendulum consists of a meterstick that
is pivoted about a small hole that is x cm from the
50.0 cm mark and observed to have a period of
oscillation of 2.50 s. What is x?
A 2500 kg wrecking ball is suspended from a 17.3
m cable. What would be its frequency of
oscillation for a small displacement?