Download Physics 121 Homework #8

Physics 121 Homework #8
Due 5 p.m., Friday, November 6
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Reading and topic coverage
Throughout this assignment, “R” refers to Six Ideas That Shaped Physics: Unit R:
The Laws of Physics Are Frame-Independent, Second Edition, by Thomas Moore.
Here is the tentative plan for upcoming classes. As always, this is subject to change.
Fri., Oct. 30
Mon., Nov. 2
Wed., Nov. 4
Fri., Nov. 6
§R8.1-8.3
§R8.4
§RB∗
§R9
Pole-in-Barn; Causality
Velocity transformation
Relativistic Doppler shift
Four-vectors revisited
Relativistic momentum & energy
Following week
Relativistic momentum & energy,
continued
∗
§RB is an appendix in the textbook. We will cover the relativistic
calculation in it, but we will present results in terms of period and
frequency rather than wavelength.
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Problems to work but not turn in.
A. Problem R8B.1.
B. Problem R8B.4.
C. A spaceship has a tail-light which blinks every 2.0 seconds.
(a) The spaceship moves away from Earth at speed 0.5. People on earth see
the tail-light as the spaceship files away. What do they see as the interval
between blinks?
(b) If they saw it blink at intervals of 5.0,s, how fast would they conclude the
spaceship was moving?
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Problems to turn in.
1. (The following are in the “two-minute problems” section at the end of the
chapter, which comes just before the main problem section.)
(a) Problem R8T.1.
(b) Problem R8T.2.
Continued on the next page....
2. (a) Problem R8S.6.
(b) In addition to answering the question asked in the above problem, draw
an accurate spacetime diagram showing the two events (in the rest frame
of the Sun and the Earth) and give a numerical value for s2 .
Notes. The Macdonald family lives on Earth. You will need to look up the
distance from the Earth to the Sun and convert it to seconds. Assume the Sun
and the Earth are at rest relative to one another.
3. Problem R8B.7.
Note. Skip the part of the problem which asks for a careful explanation of your
solution. You only need to give a numerical answer.
4. Problem R8S.10.
5. Problem R8S.11. In particular:
(a) As a warmup, show that a particle traveling travel at the speed of light
in the +x0 direction according to observers in the Other Frame (so that
vx0 = 1), is also observed in the Home Frame to be traveling at the speed
of light in the +x direction (so that vx = 1). This is true no matter what
the value of β is.
(b) Next, generalize this result to a particle traveling at the speed of light in the
Other Frame at some angle relative to the x0 direction, so that its velocity
has nonzero vx0 and vy0 components (see figure below). Write expressions
for vx0 and vy0 in terms of θ0 , use the velocity transformation formulas to
get expressions for vx and vy in terms of θ0 and β, and combine them to
show that the magnitude of velocity, v, in the Home Frame is 1.
v’
v’y
θ’
v’x
Continued on the next page....
6. Hydrogen atoms that undergo a certain transition from one energy state to
another (the “hyperfine line”) emit radio waves with frequency 1420.406 MHz
(in the rest frame of the hydrogen).
(a) The nearest big spiral galaxy to our own Milky Way is the Andromeda
Galaxy. It is moving toward us at a speed of 301 km
. At what frequency
s
is the hyperfine line from Andromeda detected on Earth?
(b) One of the receivers on the Arecibo Radio Telescope can detect radio waves
between 1150 MHz and 1730 MHz. If this receiver is used to detect the
hyperfine line in distant galaxies, what is the fastest-moving galaxy going
(i) away from us, and (ii) toward us that can be detected with this receiver?
7. * I am involved in a research project that uses radio telescopes to search for
pulsars. A pulsar is a dense, rapidly rotating neutron star which emits regular
pulses of radio waves as it rotates, like the flashes of light from a lighthouse.
The figure on the next page shows observations of a pulsar we discovered a
couple years ago, PSR J1903+0327. The pulsar is in an orbit with another star,
so the period detected at the Earth on any given day is Doppler shifted because
of the pulsar’s orbital motion. (The Earth’s motion also influences the received
period, but that effect has been removed from the plot.)
The period between pulses is 2149.91 µs as measured by someone at rest relative
to the pulsar.
These data were taken between September 2005 and January 2007 using
telescopes at Arecibo, Puerto Rico and Green Bank, West Virginia. The dots
show actual measurements, and the line shows a model of binary model inferred
from these measurements.
The x-axis shows the day of observation, with x = 0 on September 15,
2005 (Modified Julian Day number 53628 in astronomer-speak). The y-axis
shows the difference, in µs, between the observed period of the pulsar on any
given day and its source period of 2149.91 µs. For example, the dot near the
left side of the graph, at y = 0.12 µs, corresponds to an observed period of
2149.91+0.12 = 2150.03µs.
(a) Each pulse corresponds to one complete rotation of this star. If the period
between pulses is 2149.91 µs measured in the pulsar reference frame, how
many times per second does this star rotate? (If you think about your
answer, it should seem to be an unbelievably high number.)
(b) Where in the graph corresponds to times that the pulsar is moving in its
orbit most rapidly toward us (give one x value)?
(c) What about times it is moving away from us (give one x value)?
(d) What is its fastest speed while moving toward us?
(e) While moving away from us?
The figure for this problem is on the next page....
8. Here’s a puzzler left over from last week’s material.
Two spacecraft are initially at rest in the Home Frame, separated by some
distance L along the x-axis. There is a string stretched between the two
spacecraft. The string has length L, and if it is stretched much beyond this
length, it will break.
At time t = 0, the spacecraft start accelerating, that is, they start moving in the
+x direction, and their velocities slowly increase over time. The two spacecraft
undergo identical accelerations, so that observers in the Home Frame always
observe their velocities to be the same and observe the distance between the
spacecraft to be the same as the initial value, L.
What happens to the string: does it just sit there between the spacecraft,
unaffected by their acceleration? Or does it break? Or does it go slack?