Physics 207, Lecture 18, Nov' 3

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Physics 207, Lecture 18, Nov. 3

• Goals

• Chapter 14
• Interrelate the physics and mathematics of
  oscillations.
• Draw and interpret oscillatory graphs.
• Learn the concepts of phase and phase constant.
• Understand and use energy conservation in
  oscillatory systems.
• Understand the basic ideas of damping and
  resonance.

Phase Contrast Microscopy Epithelial cell in
brightfield (BF) using a 40x lens (NA 0.75)
(left) and with phase contrast using a DL Plan
Achromat 40x (NA 0.65) (right). A green
interference filter is used for both images.
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Physics 207, Lecture 18, Nov. 3

• Assignment
• HW8, Due Wednesday, Nov. 12th
• Wednesday Read through Chapter 15.4

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Periodic Motion is everywhere

• Examples of periodic motion
• Earth around the sun
• Elastic ball bouncing up an down
• Quartz crystal in your watch, computer clock,
  iPod clock, etc.

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Periodic Motion is everywhere

• Examples of periodic motion
• Heart beat
• In taking your pulse, you count 70.0 heartbeats
  in 1 min.
• What is the period, in seconds, of your heart's
  oscillations?
• Period is the time for one oscillation
• T 60 sec/ 70.0 0.86 s
• What is the frequency?
• f 1 / T 1.17 Hz

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A special kind of period oscillator Harmonic
oscillatorWhat do all harmonic oscillators
have in common?

• 1. A position of equilibrium
• 2. A restoring force, which must be linear
• Hookes law spring F -kx
• (In a pendulum the behavior only linear for
  small angles sin ? where ? s / L) In
  this limit we have F -ks with k mg/L)
• 3. Inertia
• 4. The drag forces are reasonably small

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Simple Harmonic Motion (SHM)

• In Simple Harmonic Motion the restoring force on
  the mass is linear, that is, exactly proportional
  to the displacement of the mass from rest
  position
• Hookes Law F -kx
• If k gtgt m ? rapid oscillations ltgt large
  frequency
• If k ltlt m ? slow oscillations ltgt low frequency

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Simple Harmonic Motion (SHM)

• We know that if we stretch a spring with a mass
  on the end and let it go the mass will, if there
  is no friction, .do something
• 1. Pull block to the right until x A
• 2. After the block is released from x A, it
  will
• A remain at rest
• B move to the left until it reaches
• equilibrium and stop there
• C move to the left until it reaches
• x -A and stop there
• D move to the left until it reaches
• x -A and then begin to move to
• the right

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Simple Harmonic Motion (SHM)

• We know that if we stretch a spring with a mass
  on the end and let it go the mass will .
• 1. Pull block to the right until x A
• 2. After the block is released from x A, it
  will
• A remain at rest
• B move to the left until it reaches
• equilibrium and stop there
• C move to the left until it reaches
• x -A and stop there
• D move to the left until it reaches
• x -A and then begin to move to
• the right
• This oscillation is called Simple Harmonic Motion

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Simple Harmonic Motion (SHM)

• The time it takes the block to complete one cycle
  is called the period.
• Usually, the period is denoted T and is
  measured in seconds.
• The frequency, denoted f, is the number of cycles
  that are completed per unit of time f 1 / T.
• In SI units, f is measured in inverse seconds,
  or hertz (Hz).
• If the period is doubled, the frequency is
• A. unchanged
• B. doubled
• C. halved

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Simple Harmonic Motion (SHM)

• An oscillating object takes 0.10 s to complete
  one cycle that is, its period is 0.10 s.
• What is its frequency f ?
• Express your answer in hertz.
• f 1/ T 10 Hz

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Simple Harmonic Motion

• Note in the (x,t) graph that the vertical axis
  represents the x coordinate of the oscillating
  object, and the horizontal axis represents time.
• Which points on the x axis are located a
  displacement A from the equilibrium position ?
• A. R only
• B. Q only
• C. both R and Q

Position
time
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Simple Harmonic Motion

• Suppose that the period is T.
• Which of the following points on the t axis are
  separated by the time interval T?
• A. K and L
• B. K and M
• C. K and P
• D. L and N
• E. M and P

time
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Simple Harmonic Motion

• Now assume that the t coordinate of point K is
  0.0050 s.
• What is the period T , in seconds?
• T 0.02 s
• How much time t does the block take to travel
  from the point of maximum displacement to the
  opposite point of maximum displacement
• t 0.01 s

time
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Simple Harmonic Motion

• Now assume that the x coordinate of point R is
  0.12 m.
• What distance d does the object cover during one
  period of oscillation?
• d 0.48 m
• What distance d does the object cover between the
  moments labeled K and N on the graph?
• d 0.36 m

time
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SHM Dynamics

• At any given instant we know that F ma must be
  true.
• But in this case F -k x and
  ma
• So -k x ma

a differential equation for x(t) !
Simple approach, guess a solution and see if it
works!
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SHM Solution...

• Try either cos ( ? t ) or sin ( ? t )
• Below is a drawing of A cos ( ? t )
• where A amplitude of oscillation
• with w (k/m)½ and w 2p f 2p /T
• Both sin and cosine work so need to include both

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Combining sin and cosine solutions
x(t) B cos wt C sin wt A cos
( wt f) A (cos wt cos f sin wt sin
f ) A cos f cos wt A sin f sin
wt) Notice that B A cos f C -A sin f ?
tan f -C/B
?
?
??
sin
cos
Use initial conditions to determine phase ? !
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Energy of the Spring-Mass System
We know enough to discuss the mechanical energy
of the oscillating mass on a spring.
x(t) A cos ( wt f) If x(t) is
displacement from equilibrium, then potential
energy is U(t) ½ k x(t)2 A2 cos2 ( wt
f) v(t) dx/dt ? v(t) A w (-sin ( wt
f)) And so the kinetic energy is just ½ m
v(t)2 K(t) ½ m v(t)2 (Aw)2 sin2 ( wt
f) Finally, a(t) dv/dt -?2A cos(?t ?)
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Energy of the Spring-Mass System
Kinetic energy is always K ½ mv2 ½
m(?A)2 sin2(?tf) Potential energy of a spring
is, U ½ k x2 ½ k A2 cos2(?t ?) And w2
k / m or k m w2 U ½ m w2 A2 cos2(?t
?)
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Energy of the Spring-Mass System
And the mechanical energy is K U ½ m w2 A2
cos2(?t ?) ½ m w2 A2 sin2(?t ?) K U ½
m w2 A2 cos2(?t ?) sin2(?t ?) K U ½
m w2 A2 ½ k A2 which is constant
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Energy of the Spring-Mass System
So E K U constant ½ k A2
At maximum displacement K 0 and U ½ k A2
and acceleration has it maximum (or minimum) At
the equilibrium position K ½ k A2 ½ m v2
and U 0
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SHM So Far

• The most general solution is x A cos(?t ?)
• where A amplitude
• ? (angular) frequency
• ? phase constant
• For SHM without friction,
• The frequency does not depend on the amplitude !
• This is true of all simple harmonic motion!
• The oscillation occurs around the equilibrium
  point where the force is zero!
• Energy is a constant, it transfers between
  potential and kinetic

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What about Vertical Springs?

• For a vertical spring, if y is measured from the
  equilibrium position
• Recall force of the spring is the negative
  derivative of this function
• This will be just like the horizontal case-ky
  ma

j
k
y 0
F -ky
m
Which has solution y(t) A cos( ?t ?)
where
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The Simple Pendulum (In class torques, t Ia,
were used instead but the results are the same)

• A pendulum is made by suspending a mass m at the
  end of a string of length L. Find the frequency
  of oscillation for small displacements.
• S Fy mac T mg cos(q) m v2/L
• S Fx max -mg sin(q)
• If q small then x ? L q and sin(q) ? q
• dx/dt L dq/dt
• ax d2x/dt2 L d2q/dt2
• so ax -g q L d2q / dt2 ? L d2q / dt2 - g q
  0
• and q q0 cos(wt f) or q q0 sin(wt
  f)
• with w (g/L)½

z
y
?
L
x
T
m
mg
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Velocity and Acceleration
Position x(t) A cos(?t ?) Velocity v(t)
-?A sin(?t ?) Acceleration a(t) -?2A
cos(?t ?)
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Physics 207, Lecture 18, Nov. 3

• Assignment
• HW8, Due Wednesday, Nov. 12th
• Wednesday Read through Chapter 15.4
• The rest are for Wednesday, plus damping,
  resonance and
• part of Chapter 15.

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The shaker cart

• You stand inside a small cart attached to a
  heavy-duty spring, the spring is compressed and
  released, and you shake back and forth,
  attempting to maintain your balance. Note that
  there is also a sandbag in the cart with you.
• At the instant you pass through the equilibrium
  position of the spring, you drop the sandbag out
  of the cart onto the ground.
• What effect does jettisoning the sandbag at the
  equilibrium position have on the amplitude of
  your oscillation?
• It increases the amplitude.
• It decreases the amplitude.
• It has no effect on the amplitude.
• Hint At equilibrium, both the cart and the bag
• are moving at their maximum speed. By
• dropping the bag at this point, energy
• (specifically the kinetic energy of the bag) is
• lost from the spring-cart system. Thus, both the
• elastic potential energy at maximum displacement
• and the kinetic energy at equilibrium must
  decrease

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The shaker cart

• Instead of dropping the sandbag as you pass
  through equilibrium, you decide to drop the
  sandbag when the cart is at its maximum distance
  from equilibrium.
• What effect does jettisoning the sandbag at the
  carts maximum distance from equilibrium have on
  the amplitude of your oscillation?
• It increases the amplitude.
• It decreases the amplitude.
• It has no effect on the amplitude.
• Hint Dropping the bag at maximum
• distance from equilibrium, both the cart
• and the bag are at rest. By dropping the
• bag at this point, no energy is lost from
• the spring-cart system. Therefore, both the
• elastic potential energy at maximum displacement
• and the kinetic energy at equilibrium must remain
  constant.

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The shaker cart

• What effect does jettisoning the sandbag at the
  carts maximum distance from equilibrium have on
  the maximum speed of the cart?
• It increases the maximum speed.
• It decreases the maximum speed.
• It has no effect on the maximum speed.
• Hint Dropping the bag at maximum distance
• from equilibrium, both the cart and the bag
• are at rest. By dropping the bag at this
• point, no energy is lost from the spring-cart
• system. Therefore, both the elastic
• potential energy at maximum displacement
• and the kinetic energy at equilibrium must
• remain constant.

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Exercise Simple Harmonic Motion

• A mass oscillates up down on a spring. Its
  position as a function of time is shown below.
  At which of the points shown does the mass have
  positive velocity and negative acceleration ?
• Remember velocity is slope and acceleration is
  the curvature

y(t)
(a)
(c)
t
(b)
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Example

• A mass m 2 kg on a spring oscillates with
  amplitude
• A 10 cm. At t 0 its speed is at a maximum,
  and is v2 m/s
• What is the angular frequency of oscillation ? ?
• What is the spring constant k ?
• General relationships E K U constant, w
  (k/m)½
• So at maximum speed U0 and ½ mv2 E ½ kA2
• thus k mv2/A2 2 x (2) 2/(0.1)2 800 N/m, w
  20 rad/sec

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Home Exercise Simple Harmonic Motion

• 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 recalling that w
  (g/L)½

(A) T1 T2 (B) T1 gt T2 (C) T1 lt T2
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Home ExerciseSimple Harmonic Motion

• 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.
• If you are standing, the center of mass moves
  towards the pivot point and so L is less, w is
  bigger, T2 is smaller

(A) T1 T2 (B) T1 gt T2 (C) T1 lt T2
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BTW The Rod Pendulum(not tested)

• A pendulum is made by suspending a thin rod of
  length L and mass M at one end. Find the
  frequency of oscillation for small
  displacements.
• S tz I a - r x F (L/2) mg sin(q)
• (no torque from T)
• - mL2/12 m (L/2)2 a ? L/2 mg q
• -1/3 L d2q/dt2 ½ g q

z
T
?
x
CM
L
mg
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BTW General Physical Pendulum (not tested)

• Suppose we have some arbitrarily shaped solid of
  mass M hung on a fixed axis, that we know where
  the CM is located and what the moment of inertia
  I about the axis is.
• The torque about the rotation (z) axis for small
  ? is (sin ? ? ? )
  
  
  ? -MgR sinq ? -MgR???

z-axis
R
?
x
CM
Mg
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Torsion Pendulum

• Consider an object suspended by a wire attached
  at its CM. The wire defines the rotation axis,
  and the moment of inertia I about this axis is
  known.
• The wire acts like a rotational spring.
• When the object is rotated, the wire is twisted.
  This produces a torque that opposes the
  rotation.
• In analogy with a spring, the torque produced is
  proportional to the displacement ? - k ?
  where k is the torsional spring constant
• w (k/I)½

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Home Exercise Period

• All of the following torsional pendulum bobs have
  the same mass and w (k/I)½
• Which pendulum rotates the fastest, i.e. has the
  longest period? (The wires are identical)

(A)
(B)
(C)
(D)
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Reviewing Simple Harmonic Oscillators

• Spring-mass system
• Pendula
• General physical pendulum
• Torsion pendulum

where
z-axis
x(t) A cos( ?t ?)
R
?
x
CM
Mg
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Energy in SHM

• For both the spring and the pendulum, we can
  derive the SHM solution using energy
  conservation.
• The total energy (K U) of a system undergoing
  SMH will always be constant!
• This is not surprising since there are only
  conservative forces present, hence energy is
  conserved.

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SHM and quadratic potentials

• SHM will occur whenever the potential is
  quadratic.
• For small oscillations this will be true
• For example, the potential betweenH atoms in an
  H2 molecule lookssomething like this

U
x