Name:________________________
STEM 698 HW Due Thursday, Nov. 6
Capacitor Lab Questions
1.
Recall from our experiment that the voltage across a discharging capacitor is given by
V (t )  Vinitial e

1
t
RC
where Vinitial is the voltage at time t = 0.
Suppose R has a resistance of 6.9 kΩ and C has a capacitance
of 4.0 μF. How long does it take the capacitor to discharge
to 1% of its initial voltage? (Hint. It doesn’t matter
what the initial voltage is.)
2. Measuring Capacitance The capacitor discharge law is used to measure capacitance. Here
is how. Suppose you have a capacitor of unknown capacitance. You connect in parallel with a
2000Ω resistor and charge the capacitor with a 10 V battery. You disconnect the battery and
measure the voltage after 1 second. Suppose the voltage is 4.2V. Calculate the capacitance of
the capacitor.
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Boyle’s Law Problem
3. A gas has volume 5 liters at a pressure of 2 kPa. If the volume is increased to 8 liters (while
holding its temperature constant), what is the pressure of the gas?
Power Function Modeling Problems
4. The territorial area of predatory mammals (such as lions, tigers, and cheetahs) is defined to
be its defended, or exclusive, region. Wildlife biologists studying territorial areas have found
that, across species, the territorial area T is a power function of the body weight w:
T (w)  cw1.31
a. If the body weight of the mammal is doubled, how does the territorial area change?
b. If the territorial area is halved, how are the species’ body weights related?
c. If one increases body weight by 20%, by how many percent does the territorial area
increase?
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5. The intensity of light I from a light bulb is inversely proportional to the square of the distance
from the bulb d.
a. Express this relationship as a power function.
b. For a certain light bulb, the intensity of the light is 90 watts per square meter when the
distance is 5 meters. Using this information, find the constant of proportionality in part a.
c. How far away must one be from the light bulb for the intensity to drop to 60 watts per
square meter?
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6. As mentioned in the previous problem, the intensity of light is inversely proportional to the
square of the distance to the source. You also probably know that the earth does not go in
circular orbit around the sun; its orbit is an ellipse with the sun at one of the foci:
Earth
perhelion
aphelion
Sun
This drawing is very much out of
scale and exaggerates the
eccentricity of the orbit. The earth is
much smaller and much farther
away than this drawing indicates.
109 earths would fit across the
diameter of the sun, and on this
scale (sun diameter = 3mm), the
earth would be 32 cm or a little over
a foot away from the sun.
Most people don’t know that the earth is closest to the sun in January! It turns out that at its
perihelion (point at which it is closest) in January of each year, the earth is 147.1 million km
from the sun; at aphelion (in July), the earth is 152.1 million km from the sun. By how many
percent is the intensity of the sun stronger in January than in July? Show your work.
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7. For main sequence stars, there is a relationship between the relative luminosity and mass of
the star. Relative luminosity is the ratio of the luminosity of the star to that of the sun. The mass
of a star in this context is measured in terms of solar masses. Here is a small set of data on some
main sequence stars.
Star
Mass (solar masses) Luminosity L
Spica
7.3
1050
Vega
3.1
55
Altair
1
1.1
Sun
1
1
61 Cygni A
0.17
0.002
a. Using the data above, determine a best fit power law model relating luminosity to mass.
This relationship is known as the mass-luminosity relation.
b. Kruger 60 is a main-sequence star that is about 0.11 solar masses. Using the model,
estimate the relative luminosity of Kruger 60.
c. Wolf 359 has a relative luminosity of about 0.0001. Approximately how massive is Wolf
359.
d. If one star is 3 times as massive as another, how do their luminosities compare?
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8. Ecologists have studied the relationship between the number S of species of a given
taxonomic group within a given habitat (often an island) and the area of the habitat. They
have discovered a consistent relationship: over similar habitats, S is approximately a power
function of A. and for islands the powers fall within the range 0.2 and 0.4 The following
table gives, for some islands in the West Indies, the area in square miles and the number of
species of amphibians and reptiles.
Island
Area (sq. miles) Number S of species
Cuba
44000
76
Hispaniola
29000
84
Jamaica
4200
39
Puerto Rico
3500
40
Montserrat
40
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Saba
5
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a. Find the best fit power function model for this data.
b. Is the graph of S against A concave up or concave down? Explain in practical terms what
your answer means.
c. The species-area relation for the West Indies islands can be expresses as a rule of thumb:
If one island is 10 times larger than another, then it will have ________ times as many
species.
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STEM 698 Week 9 Reading Prompt Due Thursday Nov. 6
Read the short article “Size and Shape” by Stephen Jay Gould. (excerpted from Ever Since
Darwin: Reflection in Natural History. New York: W. W. Norton, 1985) and the article”On
Being the Right Size” by J. B. S. Haldane. Choose 1) one example that Gould gives that you find
interesting and 2) a different example that Haldane gives that you find interesting and in a well
written paragraph explain in your own words how the surface area to volume ratio helps account
for the size and shape of organisms or physical objects.
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