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Chapter 15: Oscillations
COURSE THEME: NEWTON’S LAWS OF MOTION!
• Chs. 5 - 13: Methods to analyze dynamics of objects in
Translational & Rotational Motion using Newton’s Laws!
Chs. 5 & 6: Newton’s Laws using Forces (translational motion)
Chs. 7 & 8: Newton’s Laws using Energy & Work (translational motion)
Ch. 9: Newton’s Laws using Momentum (translational motion)
Chs. 10 & 11: Newton’s Laws (rotational language; rotating objects).
Ch. 14: Newton’s Laws for fluids (fluid language)
NOW
• Ch. 15: Methods to analyze the dynamics oscillating objects.
First, need to discuss (a small amount of) VIBRATIONAL LANGUAGE
Then, Newton’s Laws in Vibrational Language!
Simple Harmonic Motion
• Vibration  Oscillation = back & forth motion of an object
• Periodic motion:
Vibration or Oscillation that regularly repeats itself.
Motion of an object that regularly returns to a given position after a fixed
time interval.
• A special kind of periodic motion occurs in mechanical systems
when the force acting on the object is proportional to the position of
the object relative to some equilibrium position
– If the force is always directed toward the equilibrium position, the
motion is called simple harmonic motion
• Simplest form of periodic motion: Mass m attached to an ideal spring,
spring constant k (Hooke’s “Law”: F = -kx) moving in one dimension (x).
Contains most features of more complicated systems
 Use the mass-spring system as a prototype
for periodic oscillating systems
Spring-Mass System
• Block of mass m attached to ideal
spring of constant k. Block moves on a
frictionless horizontal surface
• When the spring is neither stretched nor
compressed, block is at the
Equilibrium Position x = 0
• Assume the force between the mass
& the spring obeys “Hooke’s Law”
Hooke’s Law: Fs = - kx
x is the displacement
Fs is the restoring force. Is always directed
toward the equilibrium position
• Therefore, it is always opposite the
displacement from equilibrium
Block Motion
Possible Motion:
(a) Block displaced right of x = 0
– Position x is positive (right)
– Restoring force Fs to the left
(b) Block is at x = 0
– Spring is neither stretched nor
compressed
– Force Fs is 0
(c) Block is displaced left of x = 0
– Position x is negative (left)
– Restoring force Fs to the right
Terminology & Notation
• Displacement = x(t)
• Velocity = v(t)
• Acceleration = a(t)  Constant!!!!!!!
 Constant acceleration equations from Ch. 2
DO NOT APPLY!!!!!!!!!!!
•
•
•
•
Amplitude = A (= maximum displacement)
One Cycle = One complete round trip
Period = T = Time for one round trip
Frequency = f = 1/T = # Cycles per second Measured in Hertz (Hz)
A and T are independent!!!
• Angular Frequency = ω = 2πf = Measured in radians/s.
• Repeat: Mass-spring system contains most
features of more complicated systems
 Use mass-spring system as a prototype for
periodic oscillating systems.
• Results we get are valid for many systems
besides this prototype system.
• ANY vibrating system with restoring force
proportional to displacement (Fs = -kx) will
exhibit simple harmonic motion (SHM) and
thus is a simple harmonic oscillator (SHO).
(whether or not it is a mass-spring system!)
Acceleration
• Force described by Hooke’s Law is the net force in
Newton’s 2nd Law
FHooke  FNewton
 The acceleration is proportional kx  ma
to the block’s displacement
x
k
• Direction of the acceleration
ax   x
m
is opposite the displacement
• An object moves with simple harmonic motion
whenever its acceleration is proportional to its
position and is oppositely directed to the
displacement from equilibrium
NOTE AGAIN!
• The acceleration is not constant!
– So, the kinematic equations can’t be applied!
From previous slide: - kx = max
– If block is released from position
x = A, initial acceleration is – kA/m
– When block passes through
equilibrium position, a = 0
– Block continues to x = - A where
acceleration is + kA/m
Block Motion
• Block continues to oscillate between –A and +A
– These are turning points of the motion
• The force is conservative, so we can use the
appropriate potential energy U = (½)kx2
• Absence of friction means the motion will
continue forever!
– Real systems are generally subject to friction, so
they do not actually oscillate forever
Simple Harmonic Motion
Mathematical Representation
• Block will undergo simple harmonic motion model
• Oscillation is along the x axis
2
d
x
k
• Acceleration
a 2  x
dt
m
k
2
• Define: w 
m
• Then a = -w2x. To find x(t), a function that satisfies this
equation is needed
– Need a function x(t) whose second derivative is the same as the
original function with a negative sign and multiplied by w2
– The sine and cosine functions meet these requirements
Simple Harmonic Motion
Graphical Representation
d 2x
k
a 2  x
dt
m
• A solution to this is
x(t) = A cos (wt + f)
• A, w, f are all constants
• A cosine curve can be used
to give physical
significance to these
constants
Definitions
x(t) = A cos (wt + f)
• A  the amplitude of the motion
– This is the maximum position of the particle in either the positive
or negative direction
w is called the angular frequency
– Units are rad/s
A & f: Determined by position & velocity at t = 0
If the particle is at x = A at t = 0, then f = 0
• The phase of the motion is (wt + f)
• x (t) is periodic and its value is the same each time
wt increases by 2p radians
Period & Frequency
• Period, T  time interval required for mass
to go through one full cycle of its motion
– Values of x & v for the mass at time t equal the values
of x & v at t + T
2p
T 
w
• Frequency, f  Inverse of period.
– Number of oscillations that mass m
undergoes per unit time interval
ƒ
1 w

T 2p
• Units: cycles per second = hertz (Hz)
Summary Period & Frequency
• Frequency & period equations can be rewritten to
solve for w. Period & frequency can also be written:
2p
w  2p ƒ 
T
m
T  2p
k
1
ƒ
2p
k
m
• Frequency& period depend only on mass m of
block & the spring constant k
• They don’t depend on the parameters of motion A, v
• The frequency is larger for a stiffer spring
(large values of k) & decreases with increasing
mass of the block
x(t), v(t), a(t) for
Simple Harmonic Motion
x(t )  A cos (wt  f )
dx
v
 w A sin(w t  f )
dt
2
d x
a  2  w 2 A cos(w t  f )
dt
• Simple harmonic motion is one-dimensional& so
directions can be denoted by + or – sign
• Obviously, simple harmonic motion is not
uniformly accelerated motion. CANNOT USE
kinematic equations from earlier.
NOTE ONE MORE TIME!
The acceleration is not constant!
That is, the 1 dimensional kinematic equations for constant
acceleration (from Ch. 2) DO NOT APPLY!!!!!!
v f  vi  at
x f  xi  vit  at
v f  vi  2ax
x f  xi  vt
2
2
1
2
THESE ARE WRONG AND WILL GIVE
YOU WRONG ANSWERS!!
2
TO EMPHASIZE THIS!
THROW THESE AWAY FOR THIS CHAPTER!!
v f  vi  at
x f  xi  vit  at
v f  vi  2ax
x f  xi  vt
2
2
1
2
2
THESE ARE WRONG AND WILL GIVE
YOU WRONG ANSWERS!!
Maximum Values of v & a
x(t) = Acos(wt + f)
v(t) = - wAsin(wt + f)
a(t) = - w2Acos(wt + f)
• Sine & cosine functions oscillate between 1, so
we can easily find the maximum values of velocity
and acceleration for an object in SHM
v max
amax
k
 wA 
A
m
k
2
w A
A
m
Graphs: General
• The graphs show (general case)
(a) Displacement x(t) as a function of time
x(t) = Acos(wt + f)
(b) Velocity v(t) as a function of time
v(t) = - wAsin(wt + f)
(c ) Acceleration a(t) as as a function of time
a(t) = - w2Acos(wt + f)
• The velocity is 90o out of phase with the
displacement and the acceleration is 180o out of
phase with the displacement
Example 1
x(t) = Acos(wt + f)
v(t) = - wAsin(wt + f)
a(t) = - w2Acos(ωt + f)
• Initial conditions at t = 0:
•
•

•
x(0) = A; v(0) = 0
This means f = 0
 x(t) = Acos(wt)
v(t) = - wAsin(wt)
Velocity reaches a maximum of
 wA at x = 0
a(t) = - w2Acos(wt)
Acceleration reaches a maximum
of  w2A at x = A
Example 2
x(t) = Acos(wt + f)
v(t) = - wAsin(wt + f)
a(t) = - w2Acos(ωt + f)
• Initial conditions at t = 0
x(0) = 0; v(0) = vi
• This means f =  p/2

x(t) = Asin(wt)
v(t) = wAcos(wt)
a(t) = - w2Asin(wt)
• The graph is shifted one-quarter
cycle to the right compared to the
graph of x (0) = A