PHY100 Homework 3 (1) (1)
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University of Toronto *
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Course
100
Subject
Astronomy
Date
Dec 6, 2023
Type
Pages
4
Uploaded by SargentOryx4011
PHY100 Homework 3 Weight 8%
1.
Watch the following video by Terence Tao on Cosmic Distance Ladder and write a
200- word report on your favorite part of the talk. The talk is available at:
https://www.youtube.com/watch?v=7ne0GArfeMs
. His slides for the talk are available
here:
https://terrytao.files.wordpress.com/2010/10/cosmic-distance-ladder2.pptx
(20 pts)
My favourite part of this talk was about the measurement of stars. I was quite
surprised by the concept of parralax. The idea that an object’s displacement can
change so drastically, simply because of a change in perspective is fascinating. For
example, if one looks at the same star six months apart, there will be a distance
separation of 2AU (300,000,000 km). This phenomena offers enough parallax to
measure all stars within 100 light years. Though this is not a very large amount, it is
still possible to use nearby stars to measure more distant ones. In addition,
spectroscopy is the study of absorption and emission of light. This can be used to
measure the colour of a nearby star as well as its apparent brightness. By plotting
such measurements on what is called the Hertzsrpung-Russell Diagram, phycisists
discovered that there was a relationship between a star’s colour and its magnitude of
brightness. It was revealed that red stars are less bright, blue and white stars are
more bright, and the sun is somewhere in the middle. By observing this curve,
scientists were able to study more distant stars. They were soon able to measure all
stars within 300,000 light years, which covers the entire milky way galaxy.
2.
Principle of inertia and gravity:
Consider a boat coasting on a quiet lake with a ball hung at the top of its mast. At a
certain time ball is let go. Plot the trajectory of the ball and the boat as seen by:
a. A person on the boat (5 pts)
The person on the boat will see the ball moving straight down. The law of inertia
states that an object in motion stays in motion and an object at rest remains at rest
unless acted upon by some external force. As a result, the person should be moving
with the boat and be sharing its velocity. This should also apply to the ball. Because
of this relative motion, the ball will look as though it is simply dropping horizontally.
b. A person on ground (10 pts)
The person on the ground will see the ball fall in a sort of curved trajectory. Unlike the
person on the boat, this person will not be moving along with the boat, but rather at
rest. It will look as though the boat, along with the ball are moving horizontally. So,
when the ball drops, the vertical velocity of the ball falling down will be added to the
horizontal velocity of the boat/ball moving along the the lake. This will make it look
like the ball is following a curved path.
c. Do both people see the ball hitting the same spot on the boat or different? (5 pts)
Both people should see the ball hitting the same spot on the boat. Though each
person has a different frame of reference, the principle of gravity justifies that these
varying perspectives do not change the effects of gravity that is acting on the ball. In
both cases, the ball's mass and distance from the ground remain the same, which
means both people will see it hit the same spot.
3.
Earth and moon:
a. What is the direction of gravitational force between earth and moon? (4 pts)
Both the earth and the moon are shown to enforce a gravitational force on each
other. The earth pulls the moon towards itself, while the moon has a small
gravitational force pulling on the earth in return.
b. Is the Moon falling towards the earth? Explain your answer. (8 pts)
Yes, the moon is technically falling towards the earth due to the earth’s immense
gravitational force acting on it. If the moon was not falling into the earth, it would no
longer be in orbit and would move in a straight path away from Earth. It is important
to note that the moon also has a large centripetal velocity, which is what keeps it
orbiting rather than being pulled in completely by the earth.
c. What would be the trajectory of earth’s moon if suddenly earth’s gravitation field
vanished? (8 pts)
Earth’s gravitational’s pull is what keeps the moon in its orbit. This circular motion
would no longer exist without gravity. As mentioned in question b), the moon would
simply continue moving in a straight line away from the Earth if there was no force
acting on it. This is due to the law of inertia.
4.
Kepler’s laws:
a. Kepler observed that planets sweep out equal area in equal time, but the speed at
which planets move at different parts of the orbit is not constant. Why is it so? Explain
your answer. (10 pts)
According to Kepler’s second law, a planet should cover the same amount of space
in the same amount of time, regardless of where it is in orbit. We can think of this by
dividing the ellipse in which a planet travels, into several triangular wedges. Each of
these wedges should have the same area. However, the dimensions of these
wedges should change depending on the planet’s distance from the sun. The further
the planet is from the sun, the more the wedge should be stretched and thinned. The
closer it is, the more the wedge should be compressed. The stretched triangle should
consequently have a smaller base while the compressed one should have a larger
base which takes up more space on the orbit’s elliptical circumference. Since the
planet has to span both these lengths in the same amount of time, it should move
slower when it is further from the sun and faster when it is close to the sun.
b. Does Earth travel fastest when it’s closest to the Sun or when it’s the farthest?
Explain using Kepler’s laws. (5 pts)
Earth travels fastest when it’s closest to the sun. This is immensely related to
question a). As mentioned previously, when Earth is closer to the sun, the “wedge”
that corresponds to its distance at this point will be more compressed and have a
larger base. It will therefore take up a larger portion of the orbit’s circumference.
Since the earth must travel a longer distance, it will have to move faster.
c. What kind (shape) of an orbit will allow Kepler’s area law to hold but now the
planets can have constant speed throughout its orbit. Justify your choice of shape?
(10 pts)
A perfect circle will allow this area law to hold while permitting constant speeds
among the planets. Since the radius between the sun and planets will always stay
the same, the planets will not need to change speed depending on their location in
the orbit. This will be easy to consider if we imagine the concept with the wedges
from question 4; in this case, all wedges will be identical in shape, so the speed will
be identical as it moves from wedge to wedge.
5. Summarize your understanding of what one means by “universality of a law” using
Newton’s law of gravitation as an example, in no more than 100 words. If you use
any resources to answer this, please cite them. (15 pts)
The law of universality simply denotes the idea that all areas of the universe are
subject to the same laws. This includes both on Earth and extraterrestrially. An
example of this is Newton's law of gravitation, which states that all objects in the
universe attract each other in proportion to their masses. This can be seen everyday,
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as gravity pulls us to the center of the earth, or in space, as the moon “falls into” the
earth while it is in orbit.
https://www.britannica.com/science/Keplers-laws-of-planetary-motion