PHYS 1303, Chap. 2, Homework, Prof. Kaim, DMC
Figure 2.21 in the textbook ("Gravity"), showing the motion of a ball near Earth's surface, __________________________________________.
depicts how gravity causes the ball to accelerate downward.
What is the maximum possible parallax of Mercury during a solar transit, as seen from either end of a 3900 km baseline on Earth? Express your answer using two significant figures.
(149,500,000 - 69,800,000)km = 79,700,000km. (Earth to Sun)-(Mercury@ Aphelion/Mercury closest to Sun)=Max Parallax Parallax: 3,900/79,700,000=0.0000489 Radian: 0.0000489 x (180/pi) degree x 3,600= arcsec = 10.0933 arcsec or 10 arc seconds
Two competing models attempt to explain the motions and changing brightness of the planets: Ptolemy's geocentric model and Copernicus' heliocentric model. Sort the characteristics according to whether they are part of the geocentric model, the heliocentric model, or both solar system models. Drag the appropriate items to their respective bins.
(Ptolemy's geocentric model was based on the idea that Earth is the center of the universe, while Copernicus's heliocentric model was developed around the idea that the Sun is at the center. While these two models were based on opposing ideas, they shared a common belief in uniform circular motion and the use of epicycles. However, Copernicus's heliocentric model does not use epicycles to explain retrograde motion like Ptolemy's geocentric model. In order to explain retrograde motion, Copernicus uses the different orbital speeds of the planets as an explanation to the seemingly backward motion of the planets in the sky.)
What is a scientific theory?
A theory is a framework of ideas and assumptions that represents our best possible explanation for something.
1. Earth is located at one focusof the Moon's orbit. 2. According to Kepler's second law, Jupiter will be traveling most slowly around the Sun when at aphelion. 3. Earth orbits in the shape of a/an ellipsearound the Sun. 4. The mathematical form of Kepler's third law measures the period in years and the __________________in astronomical units (AU). 5. According to Kepler's second law, Pluto will be traveling fastest around the Sun when at perihelion. 6. The extent to which Mars' orbit differs from a perfect circle is called its eccentricity.
1. focus 2. aphelion 3. ellipse 4. semimajor axis 5. perihelion 6. eccentricity
Halley's comet has a perihelion distance of 0.6 AU and an orbital period of 76 years. What is the aphelion distance of Halley's comet from the Sun? Express your answer using two significant figures.
76^2/3 = 17.9422 aphelion 0.6 perihelion 2(17.9422) - 0.6 = 35.2844 =35 (two significant figures)
What is the gravitational acceleration at the altitude of 5,000 km? Take Earth's radius to be 6400 km. Express your answer using two significant figures.
9.8 x (6400/11400)^2 = 3.0887 or 3.1 m/s^2 gravitational constant x (earth's radius/earth's radius + altitude)^2 = gravitational acceleration at the altitude of 5000km
What is the gravitational acceleration at the altitude of 50,000 km? Take Earth's radius to be 6400 km. Express your answer using two significant figures.
9.8 x (6400/56,400)^2 = 0.126191 or .13 m/s^2 gravitational constant x (earth's radius/earth's radius + altitude)^2 = gravitational acceleration at the altitude of 50,000km
What is the gravitational acceleration at the altitude of 500 km? Take Earth's radius to be 6400 km. Express your answer using two significant figures.
9.8 x (6400/6900)^2 = 8.43117 or 8.4 m/s^2 gravitational constant x (earth's radius/earth's radius + altitude)^2 = gravitational acceleration at the altitude of 500km
The following diagrams are the same as those from Part A. This time, rank the pairs from left to right based on the size of the acceleration the asteroid on the left would have due to the gravitational force exerted on it by the object on the right, from largest to smallest.
According to Newton's second law, the asteroid with the largest acceleration will be the one that has the strongest gravitational force exerted on it by the object on the right. That is why the ranking here is the same as the ranking for Part A.
Each of the four diagrams below represents the orbit of the same comet, but each one shows the comet passing through a different segment of its orbit around the Sun. During each segment, a line drawn from the Sun to the comet sweeps out a triangular-shaped, shaded area. Assume that all the shaded regions have exactly the same area. Rank the segments of the comet's orbit from left to right based on the length of time it takes the comet to move from Point 1 to Point 2. Rank from longest to shortest. If you think that two (or more) of the diagrams should be ranked as equal, drag one on top of the other(s) to show this equality.
All stacked in the middle. (Although Kepler wrote his laws specifically to describe the orbits of the planets around the Sun, they apply more generally. Kepler's second law tells us that as an object moves around its orbit, it sweeps out equal areas in equal times. Because all the areas shown here are equal, the time it takes the comet to travel each segment must also be the same.)
When would a new Venus be highest in the sky?
At noon (A new Venus occurs when Venus is directly between the Sun and Earth, which means a new Venus will be highest in the sky at the same time that the Sun is highest in the sky, which is around noon (local time).)
The following five diagrams show pairs of astronomical objects that are all separated by the same distanced. Assume the asteroids are all identical and relatively small, just a few kilometers across. Considering only the two objects shown in each pair, rank the strength, from strongest to weakest, of the gravitational force acting on the asteroid on the left.
Because the distance is the same for all five cases, the gravitational force depends only on the product of the masses. And because the same asteroid is on the left in all five cases, the relative strength of gravitational force depends on the mass of the object on the right. Continue to Part B to explore what happens if we instead ask about the gravitational force acting on the object on the right.
What is the escape speed from the Moon? Express your answer using two significant figures. The Moon's mass is 7.4×1022 kg and its radius is 1700km.
Formula: sq. root of 2 x (GM)/r sq. root of (2 x (6.67 x 10^-11) x (7.4 x 10^22))/(1.7 x 10^6) My figures gave the answer: v = 2409.74 m/s or v = 2410 m/s The computer accepted it as correct, but gave this as the correct answer: v = 2400 m/s
What is the speed of a spacecraft moving in a circular orbit just above the lunar surface? Express your answer using two significant figures. The Moon's mass is 7.4×1022 kg and its radius is 1700 km.
Formula: sq. root of GM/r sq. root ((6.67 x 10^-11) x (7.4 x 10^22))/(1.7 x 10^6) My figures gave the answer: v = 1704.32 m/s The computer accepted it as correct, but gave this as the correct answer: v = 1700 m/s
Consider again the diagrams from Parts A and B, which are repeated here. Again, assume that all the shaded areas have exactly the same area. This time, rank the segments of the comet's orbit based on the speed with which the comet moves when traveling from Point 1 to Point 2. Rank from fastest to slowest. If you think that two (or more) of the diagrams should be ranked as equal, drag one on top of the other(s) to show this equality.
From Parts A and B, you know that the comet takes the same time to cover each of the four segments shown, but that it travels greater distances in the segments that are closer to the Sun. Therefore, its speed must also be faster when it is closer to the Sun. In other words, the fact that that the comet sweeps out equal areas in equal times implies that its orbital speed is faster when it is nearer to the Sun and slower when it is farther away.
Each of the following diagrams shows a spaceship somewhere along the way between Earth and the Moon (not to scale); the midpoint of the distance is marked to make it easier to see how the locations compare. Assume the spaceship has the same mass throughout the trip (that is, it is not burning any fuel). Rank the five positions of the spaceship from left to right based on the strength of the gravitational force that Earth exerts on the spaceship, from strongest to weakest.
Gravity follows an inverse square law with distance, which means the force of gravity between Earth and the spaceship weakens as the spaceship gets farther from Earth.
The following diagrams are the same as those from Part A. This time, rank the five positions of the spaceship from left to right based on the strength of the gravitational force that the Moon exerts on the spaceship, from strongest to weakest.
Gravity follows an inverse square law with distance, which means the force of gravity between the Moon and the spaceship increases as the spaceship approaches the Moon. Now continue to Part C for activities that look at the effects of both distance and mass on gravity.
Suppose you are in an elevator. As the elevator starts upward, its speed will increase. During this time when the elevator is moving upward with increasing speed, your weight will be __________.
Greater than your normal weight at rest. (Increasing speed means acceleration, and when the elevator is accelerating upward you will feel a force pressing you to the floor, making your weight greater than your normal (at rest) weight.)
Copernicus's heliocentric model and Ptolemy's geocentric model were each developed to provide a description of the solar system. Both models had advantages that made each an acceptable explanation for motions in the solar system during their time. Sort each statement according to whether it is an advantage of the heliocentric model, the geocentric model, or both. Drag the appropriate items to their respective bins.
Heliocentric: Explained planetary motions and brightness changes most simply Geocentric: Rooted in widely accepted religious beliefs regarding Earth's place in the universe Both geocentric and heliocentric: Planetary orbits and motions based on Greek ideologies of perfect form and motion Predicted planetary positions accurately over relatively short time periods (The geocentric model was compelling because it adhered to religious beliefs about Earth's centrality in the universe. The heliocentric model was compelling because it provided a simpler explanation for observed motions in the solar system. Because both models adhered to the belief in perfect form and motion, they made inaccurate predictions of planetary motions over long periods of time. Since neither model made better predictions than the other, both remained viable.)
Consider again the diagrams from Parts D and E, which are repeated here. Again, imagine that you observed the asteroid as it traveled for one week, starting from each of the positions shown. This time, rank the positions (A-D) from left to right based on how fast the asteroid is moving at each position. Rank from fastest to slowest. If you think that two (or more) of the diagrams should be ranked as equal, drag one on top of the other(s) to show this equality.
Just as you found for the comet in Parts A through C, the asteroid must be traveling at a higher speed during parts of its orbit in which it is closer to the Sun than during parts of its orbit in which it is farther away. You should now see the essence of Kepler's second law: Although the precise mathematical statement tells us that an object sweeps out equal areas in equal times, the key meaning lies in the idea that an object's orbital speed is faster when nearer to the Sun and slower when farther away. This idea explains why, for example, Earth moves faster in its orbit when it is near perihelion (its closest point to the Sun) in January than it does near aphelion (its farthest point from the Sun) in July.
Johannes Kepler used decades of Tycho Brahe's observational data to formulate an accurate description of planetary motion. Kepler spent almost 30 years of his life trying to develop a simple description of planetary motion based on a heliocentric model that fit Tycho's data. What conclusion did Kepler eventually come to that revolutionized the heliocentric model of the solar system?
Kepler determined that the planetary orbits are elliptical. (One of the most crucial conclusions that Kepler reached using Tycho's data was that the planets do not move in circular orbits, but rather in elliptical orbits. Kepler also concluded that the planets do not move with uniform motion. Applying these ideas to the Copernican model, the revised heliocentric model could then accurately predict planetary positions over long periods of time.)
Each of the four diagrams below represents the orbit of the same asteroid, but each one shows it in a different position along its orbit of the Sun. Imagine that you observed the asteroid as it traveled for one week, starting from each of the positions shown. Rank the positions based on the area that would be swept out by a line drawn between the Sun and the asteroid during the one-week period. Rank from largest to smallest. If you think that two (or more) of the diagrams should be ranked as equal, drag one on top of the other(s) to show this equality.
Kepler's second law tells us that the asteroid will sweep out equal areas in equal time intervals. Therefore, the area swept out in any one week period must always be the same, regardless of the asteroid's location in its orbit around the Sun.
Consider again the diagrams from Part A, which are repeated here. Again, assume that all the shaded areas have exactly the same area. This time, rank the segments of the comet's orbit from left to right based on the distance the comet travels when moving from Point 1 to Point 2. Rank from longest to shortest. If you think that two (or more) of the diagrams should be ranked as equal, drag one on top of the other(s) to show this equality.
Kepler's second law tells us that the comet sweeps out equal areas in equal times. Because the area triangle is shorter and squatter for the segments nearer to the Sun, the distance must be greater for these segments in order for all the areas to be the same.
Each of the following diagrams shows a planet orbiting a star. Each diagram is labeled with the planet's mass (in Earth masses) and its average orbital distance (in AU). Assume that all four stars are identical. Use Kepler's third law to rank the planets from left to right based on their orbital periods, from longest to shortest. If you think that two (or more) of the diagrams should be ranked as equal, drag one on top of the other(s) to show this equality. (Distances are to scale, but planet and star sizes are not.)
Kepler's third law tells us that the orbital period of the planet depends on its average distance from its star, but not on the planet's mass. As Newton later showed with his version of Kepler's third law, this is actually an approximation that works well whenever the planet's mass is small compared to the mass of the star.
When would you expect to see Venus high in the sky at midnight?
Never. (For Venus to be high in the sky at midnight, it would have to be on the opposite side of our sky from the Sun. But that never occurs because Venus is closer than Earth to the Sun.)
The following diagrams are the same as those from Part A. Again considering only the two objects shown in each pair, this time rank the strength, from strongest to weakest, of the gravitational force acting on the object on the right.
Newton's third law tells us that the gravitational force exerted on the asteroid on the left by the object on the right will be equal in magnitude, but opposite in direction to the gravitational force exerted on the object on the right by the asteroid on the left. That is why the ranking here is the same as the ranking for Part A.
The following diagrams all show the same star, but each shows a different planet orbiting the star. The diagrams are all scaled the same. (For example, you can think of the tick marks along the line that passes through the Sun and connects the nearest and farthest points in the orbit as representing distance in astronomical units (AU).) Rank the planets from left to right based on their average orbital distance from the star, from longest to shortest. (Distances are to scale, but planet and star sizes are not.)
Note that the line that passes through the star and connects the nearest and farthest points of the planet's orbit is called the major axis, and half this line is the semimajor axis — which we consider the planet's average distance from the star.
Consider again the diagrams from Part D, which are repeated here. Again, imagine that you observed the asteroid as it traveled for one week, starting from each of the positions shown. This time, rank the positions from left to right based on the distance the asteroid will travel during a one-week period when passing through each location. Rank from longest to shortest. If you think that two (or more) of the diagrams should be ranked as equal, drag one on top of the other(s) to show this equality.
Notice the similarity between what you have found here and what you found for the comet in Part B. Kepler's second law tells us any object will sweep out equal areas in equal times as it orbits the Sun, which means the area triangles are shorter and squatter when the object is nearer to the Sun, so that the object covers a greater distance during any particular time period when it is closer to the Sun than when it is farther away.
The following diagrams are the same as those from Part A. This time, rank the planets from left to right based on the amount of time it takes each to complete one orbit, from longest to shortest. If you think that two (or more) of the diagrams should be ranked as equal, drag one on top of the other(s) to show this equality. (Distances are to scale, but planet and star sizes are not.)
Recall that the time it takes a planet to complete an orbit is called its orbital period. The pattern found in this Part illustrates one of the ideas that are part of Kepler's third law: Planets with larger average orbital distances have longer orbital periods.
The geocentric model of Aristotle had the __________, Moon, planets, and stars orbiting a stationary _______________. A modification by Ptolemy had most of the planets moving in small circles called ____________. The center of these ____________ moved around the ___________ in larger circles called ____________. Over the centuries, however, other astronomers further altered the model, and dozens of circles were needed to fully describe the motions of the 7 visible "planets."
Sun, Earth, epicycles, epicycles, Earth, deferents
How long would a radar signal take to complete a round-trip between Earth and Mars when the two planets are 0.8 AU apart?
T=d/v (0.8*150*10^6)km/300,000km/s =1.20000000km/300,000km/s =400s*2 =800s
Galileo Galilei was the first scientist to perform experiments in order to test his ideas. He was also the first astronomer to systematically observe the skies with a telescope. Galileo made four key observations that challenged the widely accepted philosophical beliefs on which the geocentric model was based, thus providing support for the heliocentric model. From the following list of observations, which are the key observations made by Galileo that challenged widespread philosophical beliefs about the solar system?
The Moon has mountains, valleys, and craters. The Sun has sunspots and rotates on its axis. Jupiter has orbiting moons. Venus goes through a full set of phases. (Galileo made four key observations that went against the geocentric model and the common beliefs about the universe at the time. Observing that the Sun and Moon had surface blemishes disproved the idea that celestial objects were perfect. Galileo's observations of Jupiter's orbiting moons showed that there were other centers of motion in the universe. Galileo's most crucial observation was the observation of Venus in different phases, which directly supported the idea that objects orbit the Sun rather than Earth.)
As you found in Part A, your weight will be greater than normal when the elevator is moving upward with increasing speed. For what other motion would your weight also be greater than your normal weight?
The elevator moves down while slowing in speed. (When the elevator is moving downward, a downward acceleration would mean an increasing downward speed. Therefore, as your answer correctly states, an upward acceleration would mean a decreasing downward speed.)
The geocentric model, in all of its complexity, survived scientific scrutiny for almost 1,400 years. However, in modern astronomy, scientists seek to explain the natural and physical world we live in as simply as possible. The complexity of Ptolemy's model was an indicator that his theory was inherently flawed. Why, then, was the geocentric model the leading theory for such a long time, even though the heliocentric model more simply explained the observed motions and brightness of the planets?
The heliocentric model did not make noticeably better predictions than the geocentric model. Ancient astronomers did not observe stellar parallax, which would have provided evidence in favor of the heliocentric model. The geocentric model conformed to both the philosophical and religious doctrines of the time. From Earth, all heavenly bodies appeared to circle around a stationary Earth. (The geocentric model was the leading model for hundreds of years because it conformed to the common beliefs from observations and religious doctrine of the time. However, the heliocentric model gained widespread acceptance when astronomers obtained new evidence after the time of Copernicus. These astronomers popularized his view and helped pave the way for our current understanding of the solar system.)
The force of gravity follows an inverse square law, meaning that the strength of the force declines with the square of the distance between two masses. But if the distances between pairs of objects are all the same, as in Part A, then the strength of gravity depends only on __________.
The product of the two objects masses (M1 x M2). (Because the distance is the same for all five object pairs shown in Part A, the gravitational force depends only on the product of the masses. Moreover, notice that the asteroid on the left has the same mass in all five cases. With these facts in mind, you should be able to answer Part A.)
Can a theory ever be proved to be absolutely true?
Theory can never be proven to be absolutely true.
Select the correct explanation of why the geocentric model was accepted for so long.
There was no sensation of motion as the Earth moved through space. There was parallax as the Earth moved around the Sun, but it was not measurable until the mid-1800s.
The following diagrams are the same as those from Parts A and B. This time, rank the planets from left to right based on their average orbital speed, from fastest to slowest. If you think that two (or more) of the diagrams should be ranked as equal, drag one on top of the other(s) to show this equality. (Distances are to scale, but planet and star sizes are not.)
This pattern illustrates another of the ideas that are part of Kepler's third law: Planets with larger average orbital distances have slower average speeds.
Astronomers have made many observations since the days of Galileo and Kepler to confirm that the Sun really is at the center of the solar system, and that the planets revolve around the Sun in elliptical orbits. Which observation(s) could you make today that Galileo and Kepler could not have made to confirm that the heliocentric model is correct?
Transit of an extrasolar planet Doppler shifts in stellar spectra of nearby stars Stellar parallax in nearby stars (When Galileo observed the changing phases of Venus, he showed that at least one planet must be orbiting the Sun. Today, astronomers are confident that all of the planets in our solar system orbit the Sun because of Earth-based observational evidence that supports a heliocentric model. Doppler shifts and stellar parallaxes show that Earth is in motion around the Sun and is thus not stationary. More recent observations of extrasolar planets show astronomers that planets in other planetary systems are orbiting stars.)
Venus's phase is new as viewed from Earth when __________.
Venus is directly between Earth and the Sun (Now ask yourself: As viewed from Earth, how far apart in the sky would you expect Venus and the Sun to appear at this time?)
The major axis for a new comet has been determined to be 100 AU. The eccentricity of the orbit was measured to be 0.94. What is the distance of closest approach to the Sun for this comet?
a/2*(1-e)= 100/2*(1-0.94)= 50*.06=3AU
In Ptolemy's Earth-centered model for the solar system, Venus always stays close to the Sun in the sky and, because it always stays between Earth and the Sun, its phases range only between new and crescent. The following statements are all true and were all observed by Galileo. Which one provides evidence that Venus orbits the Sun and not Earth?
We sometimes see gibbous (nearly but not quite full) Venus. (In the Ptolemaic system, we should never see more than a crescent in Venus. Because we do in fact see more, the Ptolemaic model must be wrong. The full range of phases that we see for Venus is consistent only with the idea that Venus orbits the Sun. Galileo was the first to observe the phases of Venus — and hence to find this evidence in support of the Sun-centered system — because he was the first to observe Venus through a telescope. Without a telescope, we cannot tell that Venus goes through phases.)
The following diagrams show five pairs of asteroids, labeled with their relative masses (M) and distances (d) between them. For example, an asteroid with M=2 has twice the mass of one with M=1 and a distance of d=2 is twice as large as a distance of d=1. Rank each pair from left to right based on the strength of the gravitational force attracting the asteroids to each other, from strongest to weakest.
You have correctly taken into account both the masses of the asteroids and the distances between them.
An accurate sketch of Jupiter's orbit around the Sun would show ________________________.
a nearly perfect circle.
Kepler's third law states that for any planet orbiting the Sun, the orbital period squared (p^2) is equal to the average orbital distance cubed (a^3), or p^2=a^3. This implies that __________.
a planet with a large average distance from the Sun has a longer orbital period than a planet with a smaller average distance from the Sun. (Now, remember that Kepler's laws apply equally well to planets orbiting stars besides the Sun, and from Part A you already know the average distance of each of the planets from their star. With this in mind, you should be able to complete Part B.)
According to Newton's second law, the greater the force exerted on an object, the greater the object's _____.
acceleration. (Newton's second law of motion, F=ma, tells us that greater force means greater acceleration. With this idea in mind, think about how your ranking for Part A is related to the ranking for Part C.Newton's second law of motion, F=ma, tells us that greater force means greater acceleration. With this idea in mind, think about how your ranking for Part A is related to the ranking for Part C.)
Imagine that Venus is in its full phase today. If we could see it, at what time would the full Venus be highest in the sky?
at noon. (Because Venus is full when it is on the opposite side of the Sun from Earth, the Sun and Venus both appear to move through the sky together at that time. Venus therefore rises with the Sun, reaches its highest point at noon, and sets with the Sun.)
A major flaw in Copernicus's model was that it still had _________________.
circular orbits.
In Ptolemy's Earth-centered model for the solar system, Venus's phase is never full as viewed from Earth because it always lies between Earth and the Sun. In reality, as Galileo first recognized, Venus is __________.
full whenever it is on the opposite side of the Sun from Earth, although we cannot see the full Venus because it is close to the Sun in the sky. (A full Venus always occurs when it is on the opposite side of the Sun as viewed from Earth. (However, we cannot see the full Venus, because it is always very close to the Sun in the sky at that time.) Galileo used this fact as evidence for the Sun-centered view of the solar system: The fact that Venus goes through all the phases must mean it goes all the way around the Sun. In contrast, in the Ptolemaic model, Venus only varies between new and crescent phases.)
An asteroid with an orbit lying entirely inside Earth's _______________.
has an orbital semimajor axis of less than 1 AU.
Suppose you are in an elevator that is moving upward. As the elevator nears the floor at which you will get off, its speed slows down. During this time when the elevator is moving upward with decreasing speed, your weight will be __________.
less than your normal weight at rest. (Even though the elevator is still moving upward, the fact that its speed is slowing means that the acceleration is downward; the situation is rather like that of a ball that is still on its way up after you throw it, even though it is being pulled downward with the acceleration of gravity. Because the acceleration of the elevator is downward, your weight is lower than normal.)
If Earth's orbit around the Sun were twice as large as it is now, the orbit would take _________________________.
more than two times longer to traverse.
Figure 2.26(b) in the textbook ("Orbits") shows the orbits of two stars of unequal masses. If one star has twice the mass of the other, then the more massive star ____________________________________________________.
moves more slowly than the less massive star.
As shown in Figure 2.12 in the textbook ("Venus Phases"), Galileo's observations of Venus demonstrated that Venus must be _________________________.
orbiting the Sun.
Planets near opposition __________________________.
rise in the East.
Consider Earth and the Moon. As you should now realize, the gravitational force that Earth exerts on the Moon is equal and opposite to that which the Moon exerts on Earth. Therefore, according to Newton's second law of motion __________.
the Moon has a larger acceleration than Earth, because it has a smaller mass.
The major axis for a particular planet is known. In order to determine the perihelion and the aphelion, what other information about the planet is needed?
the eccentricity of the orbit
If you are standing on a scale in an elevator, what exactly does the scale measure?
the force you exert on the scale. (You probably recognize that neither your mass nor the gravitational force exerted on you change when you are in an elevator. The scale measures the force that is exerted on it, which in an elevator is a combination of the force due to your normal weight and a force due to the elevator's acceleration.)
How can Newton's third law help you solve this problem?
the strength of the force that the object on the left exerts on the object on the right has to be exactly the same (but in an opposite direction) as the force the object on the right exerts on the object on the left
