Download Integrative Studies 410 Our Place in the Universe

Problem Solving
Reminder: Quantitative
Reasoning
• Amazingly powerful tool to understand the
world around us
• Fundamentals:
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Ratios
Graphs
Area &Volume
Scaling
Arithmetical statements
From Phrase to Equation
• Important skill: translate a relation into an
equation, and vice versa
• Most people have problems with this
arithmetical reasoning
Ratios
• Different types of ratios
– Fractions: 45/7 = 6.42…
• Can subtract 7 from 45 six times, rest 3
– With units: 10 ft / 100ft
• Could be a (constant) slope, e.g. for every 10ft in
horizontal direction have to go up 1 ft in vertical
direction
– Inhomogeneous ratios: $2.97/3.8 liters
Graphs
• Making a graph
– Create a table with values of the independent
variable and the function
– Draw the coordinate system on a piece of paper
– Put in (x,y) pairs
– Connect the dots
• Example: y = 3x - 1
Simple observations – profound
Questions
• Just using eyes & brain can provoke
“cosmological” questions:
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Is the Earth the center of the Universe?
How far away are Sun and Moon?
How big are they?
How big is the Earth?
How heavy is the Earth?
Earth or Sun the Center?
• Aristotle (384–322 BC)
– Argued that the planets move on spheres around the
Earth (“geocentric” model)
– Argues that the earth is spherical based on the shape of
its shadow on the moon during lunar eclipses
• Aristarchus (310–230 BC)
– Attempts to measure relative distance and sizes of sun
and moon
– Proposes, nearly 2000 years before Copernicus, that all
planets orbit the Sun, including the Earth
(“heliocentric” model)
Counter Argument
or not?
• Objection to
Aristarchus’s model:
parallax of stars is not
observed (back then)
• Aristarchus argued that
this means the stars must
be very far away
Measuring the Size of the Earth
• Eratosthenes (ca. 276 BC)
– Measures the radius of the earth to about 20%
Documentation discerns subtle Effects
Hipparchus (~190 BC)
– His star catalog a
standard reference
for sixteen centuries!
– Introduces
coordinates for the
celestial sphere
– Also discovers
precession of the
equinoxes
How far away is the Moon?
• The Greeks used a special configuration of
Earth, Moon and Sun (link) in a lunar eclipse
• Can measure EF in units of Moon’s diameter,
then use geometry and same angular size of
Earth and Moon to determine Earth-Moon
distance
That means we can size it up!
• We can then take distance (384,000 km)
and angular size (1/2 degree) to get the
Moon’s size
• D = 0.5/360*2π*384,000km = 3,350 km
How far away is the Sun?
• This is much harder to measure!
• The Greeks came up with a lower limit,
showing that the Sun is much further away
than the Moon
• Consequence: it is much bigger than the
Moon
• We know from eclipses: if the Sun is X times
bigger, it must be X times farther away
Simple, ingenious idea – hard
measurement