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Worksheet 2 - Video Worksheet |
Useful formulas
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| 1. |
Mars orbits the Sun in uniform circular motion. The radius of
Mars' orbit is 2.28 x 1011 m and its orbital speed is
2.41 x 104 m/s. |
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| a) |
Draw the free-body diagram for Mars and use it to derive an
expression for the mass of the Sun in terms of Mars'
orbital
speed, the radius of its orbit, and the universal
gravitational constant.
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| b) |
Use the expression derived in part a) to determine the mass of
the Sun.
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| 2. |
The plot below relates the orbital speed of the planets to the radius
of their orbits. |
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a)
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What is the force that keeps the planets in their orbits?
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| b) |
Why do the "outer" planets travel slower than the "inner" planets?
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| c) |
Rearrange your answer to 1a) to find the equation for the
graph above.
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| 3. |
Astronomers have studied galaxy UGC 128 for many years. They
have measured its brightness and calculated
that the mass of stars
within a radius of 1.30 x 1021 m is 3.34 x 1040 kg. Stars orbiting at
this radius has been
measured travelling at a speed of 1.30 x 105 m/s.
What percentage of the mass within this radius is dark matter?
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| 4. |
In rural Minnesota, U.S.A., there is a dark matter detector known
as the Cryogenic Dark Matter Search (CDMS)
located 700 m
underground in an abandoned mine. It involves a number of 250 g
crystals of germanium (Ge) that
are cooled down to just above
absolute zero (-273o C).
According to the weakly interacting massive particle (WIMP) theory
of dark matter, billions of WIMPs from outer
space are raining down
on Earth each second. Although they typically pass through solid
objects as if they are
not there, there is a very small chance that a
WIMP will collide with a nucleus of an atom within any material it
happens to pass through.
As a result, at CDMS there is a very small probability that a WIMP
will collide with the nucleus of a germanium atom
within the
detector, as illustrated below: |
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| a) |
In the figure above, a WIMP with a mass of 1.07 x 10-25 kg and
an initial speed of 230 km/s collides with a
stationary germanium
nucleus with a mass of 1.19 x 10-25 kg. If the WIMP is deflected
and its speed is reduced to
75 km/s, use conservation of energy
to determine how much energy is transferred to the nucleus. (It is
this energy
that scientists must somehow detect.) In calculating your answer, assume that the collision is elastic.
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| b) |
How many times smaller is this energy than the energy required
to lift a grain of sand by one millimetre
(1 x 10-7 J)?
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5. |
A friend sends you an email that expresses skepticism about the
existence of dark matter. It says: "I thought
science
was about
observation, and objects you can see? How can you say that dark
matter exists when no one
can see
it?"
Write a five to ten sentence
reply describing the evidence for dark matter and defending the
stance
that
something
does not have to be visible in order to
be understood by science. In your reply, give an example
from
everyday life of something that exists but is not visible.
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