General Physics II - Vibrations and Waves
Hooke’s Law: Fs = -kx (- = force is opposite of displacement)
The force will act such that the mass is being either pushed or pulled toward
the equilibrium position.
Simple Harmonic Motion: Net force along the direction of motion is a Hooke’s
law type force. That is when the net force is proportional to the displacement
and is in the opposite direction.
Recall:
or
or
F = ma (Newton’s second law)
F = -kx
ma = -kx
a = -kx/m
(x has its max at A {amplitude = maximum
distance traveled by an object away from
equilibrium})
Elastic Potential Energy: Energy stored in a stretched/compressed elastic device.
PEs = ½ kx2
Recall: KE = ½ mv2 and PEg = mgh
Conservation: (KE + PEg + PEs)initial = (KE+ PEg + PEs)final
Non Conservative: WNC = (KE+ PEg + PEs)final - (KE + PEg + Pes)initial
Recall: W = F d
(Constant applied force...friction is an example)
Velocity vs. Position:
E total = KE + PEs
½ kA2 = ½ mv2 + ½ kx2
or
v = ± [k/m (A2 - x2)]1/2
(Where are the maxima and minima of the velocity and acceleration?)
SHM vs. Circular Motion: The projected shadow of a mass moving with circular motion
appears to exhibit SHM.
The distance of one complete revolution = 2 A
Total time to complete one revolution (period) = T
Velocity of mass along circular orbit = v
In the case of a spring, the motion converts KE to PEs from equilibrium to its
maximum amplitude: ½ kA2 = ½ mv2
or T = 2 (m/k)½ = Time to complete one cycle of the motion
or f = 1/2 (k/m)½ = Frequency of the motion [Where f = 1/T]
Position vs. Time: Each position of the rotating mass can be described in polar coordinates.
Recall: x = A cos  and y = A sin 
Additionally:  = t
Thus:
x = A cos ( + ) = A cos (2 ft + ) ---- time
y = A sin (t + ) = A sin (2 ft + ) ------ time
y = A sin (kx) = A sin (2/ * x) ------------- position
Where,  is the phase angle...the shift of the wave from equilibrium
(+ = left and - = right)
The Pendulum:
s=L
(for small angles)
And thus: F = -kx = -ks
Ft = -mg sin 
= -mg 
(for small angles)
Therefore: Ft = -(mg / L) s ; mg / L = k
Tspring = 2 (m/k)½ ......thus......Tpendulum = 2 (L/g)½
How does an oscillating spring compare to an oscillating pendulum?
Damped Oscillations: Systems discussed thus far will oscillate infinitely due to the
absence of friction. Friction damps the motion of an oscillation.
Under Damped: Vibrating motion is preserved but the amplitude slowly
decreases until it stops.
Critically Damped: Mass returns rapidly to equilibrium after rebound and does
not oscillate.
Over Damped: The mass returns to equilibrium without passing through the
equilibrium point. Time required is greater than for critical.
Wave Motion: Waves are produced by vibrations.
What is a wave? It is the motion of a disturbance...not the material!
Mechanical waves require three things:
1. A source of the disturbance
2. A disturbable medium
3. Some physical connection through which adjacent portions of the
medium can influence each other.
All waves carry energy and momentum.
Types of Waves:
Transverse wave: Each portion of the medium that is disturbed moves
perpendicular to the wave motion.
Longitudinal wave: Each portion of the medium that is disturbed moves
parallel to the wave motion.
 = wavelength
A = amplitude
T = period
v = =x / t = λ / T = f 
Wave Speeds on Strings:
Depends on the tension of the string. More tension causes the wave to accelerate
and return to equilibrium.
v = (F / )½ ;  = mass density = mass per unit length
Superposition and Interference:
Superposition: If two or more traveling waves are moving through a medium, the
resultant wave is found by adding together the displacements of
the individual waves, point-to-point.
Two waves can pass through each other with out being destroyed or altered.
Interference: Constructive = crest meets crest and trough meets trough.
The waves are said to be “in phase”
Destructive = crest meets trough.
The waves are said to be “out of phase”
Reflection of Waves: When a wave strikes a surface it is reflected in the opposite direction.
* The reflected wave is inverted if the end of the string is fixed to the wall.
* The reflected wave is not inverted if the end of the string is free to move along the wall.
Standing Waves: Waves that hold the same pattern. All points oscillate with the same
frequency except the nodes that do not oscillate at all.
For the case of one antinode (n = 1):
L = 1 / 2
f1 = v / 1 = v / 2L = 1/2L (F / )½
(the first harmonic)
In general: fn = n f1 = n/2L (F / )½
Forced Vibration: A system that vibrates with a fixed driving frequency.
Resonant Frequency: Natural frequency of the system at maximum amplitude that is
reached by the forced driver frequency causes the system to
resonate.