Chapter 9 Oscillatory Motion

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Chapter 9 Oscillatory Motion
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9-1 The Kinematics of Simple Harmonic Motion
Any motion that repeats itself at regular
intervals is called periodic motion
Examples circular motion, oscillatory motion
This oscillation is called Simple Harmonic Motion
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The position of the object is

• angular frequency ? determined by the inertia
  of the moving objects and the restoring force
  acting on it .

SI rad/s

• amplitude A The maximum distance of
• displacement to the equilibrium point

• phase wtf, phase angle (constant) f

The value of A and f depend on the displacement
and velocity of the particle at time t 0 (the
initial conditions)
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Period T the time for one complete oscillation
(or cycle)
Frequency f number of oscillations that are
completed each second.
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The red curve differs from the blue curve
(a) only in that its amplitude is greater
(b) only in that its period is T T/2
(c) only in that f -p/4 rad rather than zero
a phase difference
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They are in phase
They have a phase difference of p
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Relations Among Position, Velocity, and
Acceleration in Simple Harmonic Motion
We can take derivatives to find velocity and
acceleration
v(t) leads x(t) by p/2
v(t) is phase shifted to the left from x(t) by
p/2
x(t) lags behind v(t) by p/2
x(t) is phase shifted to the right from v(t) by
-p/2
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a(t) is phase shifted to the left from x(t) by p
In SHM, the acceleration is proportional to the
displacement but opposite in sign, and the two
quantities are related by the square of the
angular frequency.
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9-2 A Connection to Circular Motion
A reference particle P moving in a reference
circle of radius A with steady angular velocity w
. Its projection P on the x axis executes simple
harmonic motion.
Simple harmonic motion is the projection of
uniform circular motion on a diameter of the
circle in which the latter motion occurs.
demo
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• ACT A mass oscillates up down on a spring. Its
  position as a function of time is shown below.
  Write down the displacement as the function of
  time

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9-3 Springs and Simple Harmonic Motion
The blockspring system forms a linear simple
harmonic oscillator
Combining with Newtons second law
a differential equation for x(t)
Simple harmonic motion is the motion executed by
a particle of mass m subject to a force that is
proportional to the displacement of the particle
but opposite in sign ( a restoring force).
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The period of the motion is independent of the
amplitude
t0xx0,vv0
The initial conditions
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• Example A block whose mass m is 680 g is
  fastened to a spring whose spring constant k is
  65 N/m. The block is pulled a distance x 11 cm
  from its equilibrium position at x 0 on a
  frictionless surface and released from rest at t
  0. (a)  What are the angular frequency, the
  frequency, and the period of the resulting
  motion? (b)  What is the amplitude of the
  oscillation? (c)  What is the maximum speed vm of
  the oscillating block, and where is the block
  when it occurs? (d)  What is the magnitude am of
  the maximum acceleration of the block? (e)  What
  is the phase constant f for the motion? (f)  What
  is the displacement function x(t) for the
  springblock system?

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Solution
At equilibrium point
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• Example At t 0, the displacement x(0) of the
  block in a linear oscillator is -8.50 cm. The
  block's velocity v(0) then is -0.920 m/s, and its
  acceleration a(0) is 47.0 m/s2. (a)  What is the
  angular frequency w of this system? (b)  What are
  the phase constant f and amplitude A?

Solution
Correct phase constant is1550
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Additional Constant Forces
Solution
Simple harmonic motion with the same frequency,
but equilibrium point is shifted from x0 to xx1
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Vertical Springs
Choose the origin at equilibrium position
Simple harmonic motion with equilibrium point at
y0
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• ACTA mass hanging from a vertical spring is
  lifted a distance d above equilibrium and
  released at t 0. Which of the following
  describes its velocity and acceleration as a
  function of time?

(a) v(t) -vmax sin(wt) a(t) -amax
cos(wt)
(b) v(t) vmax sin(wt) a(t) amax
cos(wt)
k
y
(c) v(t) vmax cos(wt) a(t) -amax
cos(wt)
d
t 0
0
(both vmax and amax are positive numbers)
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9-4 Energy and Simple Harmonic Motion
This is not surprising since there are only
conservative forces present, hence the total
energy is conserved.
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(a)Potential energy U(t), kinetic energy K(t),
and mechanical energy E as functions of time t
for a linear harmonic oscillator. They are all
positive. U(t) and K(t) peak twice during every
period
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Note

• The potential energy and the kinetic energy
• peak twice during every period

• The mechanical energy is conserved for a linear
• harmonic oscillator

• The dependence of energy on the square of the
• amplitude is typical of Simple Harmonic Motion

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• ACT In Case 1 a mass on a spring oscillates back
  and forth. In Case 2, the mass is doubled but the
  spring and the amplitude of the oscillation is
  the same as in Case 1. In which case is the
  maximum potential energy of the mass and spring
  the biggest?
• A. Case 1B. Case 2C. Same

Look at time of maximum displacement x A
Energy ½ k A2 0 Same for both!
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Its Not Just About Springs
Besides springs, there are many other systems
that exhibit simple harmonic motion. Here are
some examples
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Almost all systems that are in stable equilibrium
exhibit simple harmonic motion when they depart
slightly from their equilibrium position
For example, the potential between H atoms in an
H2 molecule looks something like this
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If we do a Taylor expansion of this function
about the minimum, we find that for small
displacements, the potential is quadratic
Restoring force
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Identifying SHM
c, c positive constant
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Transport Tunnel
A straight tunnel with a frictionless interior is
dug through the Earth. A student jumps into the
hole at noon. What time does he get back?
g 9.81 m/s2 RE 6.38 x 106 m
He gets back 84 minutes later, at 124 p.m.
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• Strange but true The period of oscillation does
  not depend on the length of the tunnel. Any
  straight tunnel gives the same answer, as long as
  it is frictionless and the density of the Earth
  is constant.

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Solution
(a) The equilibrium separation occurs where the
potential energy is a minimum, so we set

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(b) We do a Taylor expansion of this U(r)
function about the equilibrium separation
This has the form of the elastic potential
energy, so the motion will be simple harmonic
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9-5 The Simple Pendulum
a simple pendulum consists of a pointlike mass m
(called the bob of the pendulum) suspended from
one end of an unstretchable, massless string of
length l that is fixed at the other end
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The motion of a simple pendulum swinging through
only small angles is approximately SHM.
The period of small-amplitude pendulum is
independent of the amplitude --- the pendulum
clock
The horizontal displacement
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The energy of a simple pendulum
The total energy is conserved
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• ACT

• You are sitting on a swing. A friend gives you a
  small push and you start swinging back forth
  with period T1.
• Suppose you were standing on the swing rather
  than sitting. When given a small push you start
  swinging back forth with period T2. Which of
  the following is true

(a) T1 T2 (b) T1 gt T2 (c) T1 lt T2
You make a pendulum shorter, it oscillates faster
(smaller period)
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• ACT A pendulum is hanging vertically from the
  ceiling of an elevator. Initially the elevator
  is at rest and the period of the pendulum is T.
  Now the pendulum accelerates upward. The period
  of the pendulum will now be
• 1. greater than T
• 2. equal to T
• 3. less than T

Effective g is larger when accelerating
upward (you feel heavier)
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9-6 More About Pendulums
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The period of a physical pendulum is independent
of its total massonly how the mass is
distributed matters
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• ACT A pendulum is made by hanging a thin
• hoola-hoop of diameter D on a small nail.
• What is the angular frequency of oscillation of
  the
• hoop for small displacements? (ICM mR2 for a
  hoop)

(a) (b) (c)
pivot (nail)
D
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• Example In Figure below, a meter stick swings
  about a pivot point at one end, at distance h
  from its center of mass. (a)  What is its period
  of oscillation T? (b)  What is the distance L0
  between the pivot point O of the stick and the
  center of oscillation of the stick?

Solution
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• Example In Figure below , a penguin (obviously
  skilled in aquatic sports) dives from a uniform
  board that is hinged at the left and attached to
  a spring at the right. The board has length L
  2.0 m and mass m 12 kg the spring constant k
  is 1300 N/m. When the penguin dives, it leaves
  the board and spring oscillating with a small
  amplitude. Assume that the board is stiff enough
  not to bend, and find the period T of the
  oscillations.

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Solution
Choose the equilibrium position as the origin
T is independent of the boards length
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• Example A block of mass m is attached to a
  spring of constant k through a disk of mass M
  which is free to rotate about its fixed axis.
  Find the period of small oscillations

M
Solution
Choose the equilibrium position as the origin
T
T
T
T
m
k
mg
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9-7 Damped Harmonic Motion
A pendulum does not go on swinging forever.
Energy is gradually lost (because of air
resistance) and the oscillations die away. This
effect is called damping.
Then the equation of motion is
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If a is small
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The larger the value of t ,the slower the
exponential
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As b increases, w decreases
Some systems have so much damping that no real
oscillations occur. The minimum damping needed
for this is called critical damping
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critical damping
critical damping
heavy damping
Over (heavy) damping
light damping
(light) damping
The time of the critical damping takes for the
displacement to settle to zero is a minimum
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• Example For the damped oscillator m 250 g, k
  85 N/m, and b 70 g/s. (a)  What is the period
  of the motion? (b)  How long does it take for the
  amplitude of the damped oscillations to drop to
  half its initial value? (c)  How long does it
  take for the mechanical energy to drop to
  one-half its initial value?

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Solution
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9-8 Driven Harmonic Motion
In damped harmonic motion, a mechanism such as
friction dissipates or reduces the energy of an
oscillating system, with the result that the
amplitude of the motion decreases in time.
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After long times
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The condition for the maximum of A
If b0
In the absence of damping, if the frequency of
the force matches the natural frequency of the
system , then the amplitude of the oscillation
reaches a maximum. This effect is called resonance
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For small b, the total width at half maximum
peak becomes broader as b increases
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The role played by the frequency of a driving
force is a critical one. The matching of this
frequency with a natural frequency of vibration
allows even a relatively weak force to produce a
large amplitude vibration
Examples

• Breaking glass

• The collapse of the Tacoma Narrows Bridge

Turbulent winds set up standing waves in the
Tacoma Narrows suspension bridge leading to its
collapse on November 7, 1940, just four months
after it had been opened for traffic
demo
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Summary of chapter 9

• Simple Harmonic Motion

• Springs

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Summary of chapter 9 Cont.

• Energy

• Simple and physical pendulums

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Summary of chapter 9 Cont.

• Damped and driven harmonic motion

If b0
resonance