Download Physics 132 Prof. Buehrle 4/01/14

Quiz 4 Solutions
1. (3 pts) Two charged parallel plates are used in many devices like a mass
spectrometer or a cyclotron to speed up ions or elementary particles. The two parallel
plates are raised to different voltages, which results in an electric field between the
plates that can be treated as uniform (constant and all in the same direction). Charged
ions drift through a small hole in one of the plates and are accelerated by the field
between the plates. The figure on the right shows a C14+ ion about to drift into the region
between the plates. As it is accelerated, it will pass through the points marked a and b.
At which point will the magnitudes of the electric potential, the force felt by the ion, and the
acceleration of the ion be greater? Check one answer from each column. (The plates are close
enough together that the holes through which the ions enter and leave are small enough that
their effect on the field may be ignored.)
Potential
Force
Acceleration
____ Va > Vb
____ Fa > Fb
____ aa > ab
____ Va = Vb
____ Fa = Fb
____ aa = ab
____ Va < Vb
____ Fa < Fb
____ aa < ab
____ Not enough info to tell
____ Not enough info to tell
____ Not enough info to tell
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1
Quiz 4 Solutions
2. (3 pts) In the figure at the right are shown four arrangements of
charge. Each charge has the same magnitude, but some are + and
some are -. All distances are to the same scale. If the same test charge
were to be placed at the point P in each of the examples, in the box
below rank the order of the electric potentials the test charge would
measure from greatest to smallest. Use “>” to mean greater than and
“=” to mean equal. Do not use “<” signs. Your answer should be a
string of letters that looks something like E = F > G > H meaning E
and F are equal and bigger than G and G is bigger than H. You, of
course, should use the letters ABCD and the appropriate ranking.
C>A>B=D
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2
Quiz 4 Solutions
3. Four charges of equal magnitude are placed on a grid as shown in
the figure at the right.
3.1 (2 pts) If a small test charge were placed at the black dot
indicated on the
x-axis, in what direction would the electric field it detects point?
A. In the +x direction
E. In the +y direction
B. In the –x direction
F. In the –y direction
C. It would be 0.
G. In some other direction
D. You can’t tell without knowing the sign of the test charge
3.2 (2 pts) In the box at the right, sketch a graph of the
electrostatic potential the test charge would measure as it
moves along the x-axis. The positions of the charges are
marked on the graph by vertical bars, and the V axis crosses
the x-axis at x = 0.
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Physics 132
3
Damped Oscillations &
Resonance
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4
The Simple Pendulum
 Consider a mass m attached to a
string of length L which is free to
swing back and forth.
 If it is displaced from its lowest
position by an angle θ, Newton’s
second law for the tangential
component of gravity, parallel
to the motion, is:
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5
The Simple Pendulum
If we restrict the pendulum’s
oscillations to small angles (< 10°),
then we may use the small angle
approximation sin θ ≈ θ, where θ
is measured in radians.
and the angular frequency of the motion is found to be:
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Physics 132
6Slide 14-74
A ball on a massless, rigid rod oscillates as a simple
pendulum with a period of 2.0 s. If the ball is
replaced with another ball having twice the mass,
the period will be
Physics 132
s
4.
0
s.
2.
8
1.
0
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12%
2%
s.
E.
15%
s.
D.
33%
2.
0
C.
38%
s.
B.
1.0 s.
1.4 s.
2.0 s.
2.8 s.
4.0 s
1.
4
A.
7
On Planet X, a ball on a massless, rigid rod oscillates
as a simple pendulum with a period of 2.0 s. If the
pendulum is taken to the moon of Planet X, where
the free-fall acceleration g is half as big, the period
will be
Physics 132
s
4.
0
2.
8
1.
0
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s.
4%
s.
E.
14%
s.
D.
23%
2.
0
C.
30%
29%
s.
B.
1.0 s.
1.4 s.
2.0 s.
2.8 s.
4.0 s
1.
4
A.
8
Tactics: Identifying and Analyzing Simple
Harmonic Motion
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9
The Physical Pendulum
 Any solid object that swings
back and forth under the
influence of gravity can be
modeled as a physical pendulum.
 The gravitational torque for
small angles (θ < 10°) is:
 Plugging this into Newton’s second law for rotational
motion, τ = Iα, we find the equation for SHM, with:
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10
A Swinging Leg as a Pendulum
Whiteboard, TA
& LA
1
𝐼𝐼 = 𝑀𝑀𝐿𝐿2
3
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11
A Swinging Leg as a Pendulum
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A solid disk and a circular hoop
have the same radius and the
same mass. Each can swing back
and forth as a pendulum from a
pivot at one edge. Which has the
larger period of oscillation?
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Physics 132
di
sk
.
ci
rc
ul
bo
ar
th
ho
ha
op
ve
.
th
Th
e
s
er
am
e
is
e
pe
no
...
te
no
ug
h
in
fo
rm
...
lid
Th
ey
D.
31%
0%
so
C.
37%
31%
Th
e
B.
The solid disk.
The circular hoop.
They both have the same period.
There is not enough information
to tell.
Th
e
A.
13
Damped Oscillations
 An oscillation that runs down
and stops is called a damped
oscillation.
 One possible reason for
dissipation of energy is
the drag force due to air
resistance.
 The forces involved in dissipation
are complex, but a simple linear
drag model is:
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Physics 132
The shock absorbers in cars and trucks are
heavily damped springs. The vehicle’s vertical
motion, after hitting a rock or a pothole, is a
damped oscillation.
14
Damped Oscillations
When a mass on a spring experiences
the force of the spring as given
by Hooke’s Law, as well as
a linear drag force of
magnitude |D| = bv, the
solution is:
where the angular
frequency is given by:
Here
is the angular frequency of the
undamped oscillator (b = 0).
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Damped Oscillations
Position-versus-time graph for a damped oscillator.
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16
Damped Oscillations
 A damped oscillator has position x = xmaxcos(ωt + φ0),
where:
 This slowly changing function
xmax provides a border to the
rapid oscillations, and is called
the envelope.
 The figure shows several
oscillation envelopes,
corresponding to different
values of the damping
constant b.
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17
Energy in Damped Systems
 Because of the drag force, the mechanical energy of a
damped system is no longer conserved.
 At any particular time we can compute the mechanical
energy from:
 Where the decay constant of
this function is called the
time constant τ, defined as:
 The oscillator’s mechanical
energy decays exponentially
with time constant τ.
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Driven Oscillations and Resonance
 Consider an oscillating system that, when left to itself,
oscillates at a natural frequency f0.
 Suppose that this system is subjected to a periodic
external force of driving frequency fext.
 The amplitude of oscillations
is generally not very high
if fext differs much from f0.
 As fext gets closer and closer
to f0, the amplitude of the
oscillation rises dramatically.
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A singer or musical instrument can shatter a crystal
goblet by matching the goblet’s natural oscillation
Physics 132 frequency.
19
Driven Oscillations and Resonance
The response
curve shows the
amplitude of a
driven oscillator at
frequencies near
its natural
frequency of 2.0 Hz.
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20
Driven Oscillations and Resonance
 The figure shows the
same oscillator with
three different values
of the damping constant.
 The resonance amplitude
becomes higher and
narrower as the damping
constant decreases.
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21
The graph shows how three oscillators respond as the
frequency of a driving force is varied. If each oscillator is
started and then left alone, which will oscillate for the longest
time?
A.
B.
C.
D.
The red oscillator.
The blue oscillator.
The green oscillator.
They all oscillate for the same length of time.
66%
m
...
sa
Th
ey
al
l
os
cil
l
at
e
gr
ee
n
os
fo
rt
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ill
at
or
.
os
c
Th
e
bl
ue
Th
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cil
l
at
or
.
12% 13% 9%
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22

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https://www.youtube.com/watch
?v=xox9BVSu7Ok
Physics 132
23
General Principles
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General Principles
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Important Concepts
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26
Quiz 6 Results
35
30
25
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
10
AVG: 6.42
STDEV: 1.66
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27
Quiz 6 Solutions
1. (3 pts) Consider a single charged particle, q, that is
moving through the resistor as a part of a constant steady
current. On the average, the charge moves through the
resistor at a constant velocity. Which of the following
statements are true while the charge is moving through
the resistor?
A. There is a net force acting on the charge.
B. The net force acting on the charge is 0.
C. There is a non-zero electric force on the charge.
D. We can’t say anything without knowing more
information.
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Quiz 6 Solutions
2.1 (2 pts) A mass hanging from a spring is oscillating up and down. A
graph of its height above the ground is shown at the right. The origin of this
graph (y = 0, t = 0) is where the axes cross. The position of the mass at a
particular instant of time is marked with the letter P. At the instant marked
P, the force the spring exerts on the mass
A.
B.
C.
D.
is upward
is downward
is zero
cannot be determined from the information given.
2.2. (2 pts) The period, T, of the data shown in problem 2 is the amount of
time between the time between two successive peaks. If we drew a velocity
curve, we would find that the period of that curve
A.
B.
C.
D.
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is the same as the period of the figure in problem 2.1.
is longer than the period of the figure in problem 2.1.
is shorter than the period of the figure in problem 2.1.
cannot be determined from the information given.
Physics 132
29
Quiz 6 Solutions
3. (3 pts) Consider the inside and outside of a cell, each filled with different
concentration of NaCl and separated by a membrane. The membrane has
only one type of Ion channel that lets through only Na+. The resting Nernst
potential of the system is -100mV. This is calculated from the equation
shown at the right. Which of the following statements are true when the
system is at the resting potential?
k BT  c2 
∆V = ln  
q
c

1 
A. Some Na accumulates on the membrane on the side with higher Na
+
concentration
B. Some Na accumulates on the membrane on the side with lower Na+
concentration
C. No Na+ accumulates on either side of the membrane.
D. Some Cl- accumulates on the membrane on the side with higher Na+
concentration
E. Some Cl- accumulates on the membrane on the side with lower Na+
concentration
F. No Cl- accumulates on either side of the membrane
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Displacements on
an elastic string / spring
Each bit of the string can move up or down
(perpendicular to its length) – transverse
waves
 Each bit of string can also move
toward/away along the string length if the
string is elastic (most notable on very
deformable strings such as slinky, rubber
band). – longitundinal waves

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How do the beads move?
Whiteboard, TA
& LA
Pulse moving to the right
y
x
• Sketch the y position of the bead indicated by
the arrow as a function of time
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32
Whiteboard, TA
& LA
Describing the motion of the beads
• Sketch the velocity of each bead in
the top figure at the time shown
Pulse moving to the right
y
x
vy
x
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33
A pulse is started on the string moving to the
right. At a time t0 a photograph of the string
would look like figure 1 below. A point on the string
to the right of the pulse is marked by a spot of paint.
(x is horizontal and right, y is vertical and up)
46%
31%
A.
B.
C.
D.
E.
F.
G.
9%
1
2
3
4
5
6
7
7 None of these
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34
3%
7
5%
6
4%
5
4
3
2
3%
1
Which graph would look most like a graph of the
y displacement of the spot as a function of time?
A pulse is started on the string moving to the
right. At a time t0 a photograph of the string
would look like figure 1 below. A point on the string
to the right of the pulse is marked by a spot of paint.
(x is horizontal and right, y is vertical and up)
53%
A.
B.
C.
D.
E.
F.
G.
24%
12%
6%
1
2
3
4
5
6
7
7 None of these
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35
6
7
2%
1%
5
4
3
2%
2
1
Which graph would look most like a graph of the
x velocity of the spot as a function of time?
A pulse is started on the string moving to the
right. At a time t0 a photograph of the string
would look like figure 1 below. A point on the string
to the right of the pulse is marked by a spot of paint.
(x is horizontal and right, y is vertical and up)
70%
A.
B.
C.
D.
E.
F.
G.
18%
1
2
3
4
5
6
7
7 None of these
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36
3%
0%
7
2%
5
4
3
1%
6
6%
2
1
Which graph would look most like a graph of the
y velocity of the spot as a function of time?
A pulse is started on the string moving to the
right. At a time t0 a photograph of the string
would look like figure 1 below. A point on the string
to the right of the pulse is marked by a spot of paint.
(x is horizontal and right, y is vertical and up)
39%
23%
11%
A.
B.
C.
D.
E.
F.
G.
10%
1
2
3
4
5
6
7
7 None of these
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37
7
7%
6
5
5%
4
3
5%
2
1
Which graph would look most like a graph of the
y force of the spot as a function of time?
What controls the widths of the
pulses in time and space?
y
t
∆t
y
x
ΔL
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38
Width of a pulse


The amount of time the demonstrator’s hand
was displaced up and down determines the time
width of the t-pulse, ∆t.
The speed of the signal propagation on the string
controls the width of the x-pulse, ∆L.
– The leading edge takes off with some speed, v0.
– The pulse is over when the trailing edge is done.
– The width is determined by “how far the leading edge
got to” before the displacement was over.
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39
What Controls the Speed of the Pulse
on a Spring?
To make the pulse go to the wall faster
2.
3.
4.
5.
6.
7.
8.
9.
Move your hand up and down more quickly (but by the same
amount).
Move your hand up and down more slowly (but by the same
amount).
Move your hand up and down a larger distance in the same
time.
Move your hand up and down a smaller distance in the same
time.
Use a heavier string of the same length under the same tension.
Use a string of the same density but decrease the tension.
Use a string of the same density but increase the tension.
Put more force into the wave.
Put less force into the wave
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Physics 132
31%
21%
13%
12%
8%
7%
5%
2%
1%
M
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M e yo an
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1.
40
Speed of a bead

The speed the bead moves depends on how fast
the pulse is moving and how far it needs to travel
to stay on the string.
dy = how far bead
moves in time dt
slope
of pulse
speed
of bead
speed
of pulse
dx = how far pulse
moves in time dt
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41
Foothold principles:
Mechanical waves




Key concept: We have to distinguish the motion
of the bits of matter and the motion of the pattern.
Mechanism: the pulse propagates by each bit of string
pulling on the next.
Pattern speed: a disturbance moves into a medium
with a speed that depends on the properties of the
medium (but not on the shape of the disturbance)
Matter speed: the speed of the bits of matter depend on
both the Amplitude and shape of the pulse and pattern
speed.
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42
Foothold principles:
Mechanical waves



Key concept: We have to distinguish the motion
of the bits of matter and the motion of the pattern.
Mechanism: the pulse propagates by each bit of string
pulling on the next.
Pattern speed: a disturbance moves into a medium
with a speed that depends on the properties of the
medium (but not on the shape of the disturbance)
v0 = speed of pulse
T = tension of spring
μ = mass density of spring (M/L)

Matter speed: the speed of the bits of matter depend on
both the size and shape of the pulse and pattern speed.
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Physics 132
43
Which goes
with which?
Graph I
Graph II
Graph III
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44
The math


We express the position of a bit of string at a
particular time by labeling which bit of string by
its x position, at x at time t the position of the
string is y(x,t).
Since subtracting a d from the argument of a
function (
) shifts the graph of the
function to the right by an amount d, if we want
to set the graph of a shape f(x) into motion at a
constant speed, we just need to set d = v0t and
take
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Physics 132
45
How do waves combine?
We know how one wave moves.
What happens when we get two waves
on top of each other?
?
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46
What happens
when they overlap perfectly?
Whiteboard, TA
& LA
?
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Physics 132
47
?
50%
1.
2.
1.
2
1
3.
2.
3.
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Other
Physics 132
48
1
2
3
14%
3
35%
What happens after the waves
collide?
47%
40%
(Bounce off)
1.
2.
3.
13%
3.
(Cancel)
4.4/7/2014 Other
Physics 132
3
(Pass through)
2
2.
1
4.
49
1
2
3
4
1%
4
1.
Whiteboard, TA
& LA
How about on the same side?
?
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50
The math


We express the position of a bit of string at a
particular time by labeling which bit of string by
its x position, at x at time t the position of the
string is y(x,t).
Since subtracting a d from the argument of a
function (
) shifts the graph of the
function to the right by an amount d, if we want
to set the graph of a shape f(x) into motion at a
constant speed, we just need to set d = v0t and
take
4/7/2014
Physics 132
51