21st Century Astronomy

Laura Kay, Stacy Palen, Brad Smith

Chapter 4

Gravity and Orbits - all with Video Answers

Educators

Chapter Questions

Problem 1

If you drop two balls of the same radius, they will accelerate toward the ground. If one ball is more massive than another, its acceleration
a. will be larger than that of the less massive ball.
b. will be smaller than that of the less massive ball.
c. will be the same as that of the less massive ball.
d. will depend on other factors.

Rodger Claar

Rodger Claar

Numerade Educator

Problem 2

The force with which you pull upward on Earth is
a. zero.
b. equal to your weight.
c. equal to your mass.
d. equal to Earth's weight.

Rodger Claar

Numerade Educator

Problem 3

In Newton's universal law of gravity, the force is
a. proportional to both masses.
b. proportional to the radius.
c. proportional to the radius squared.
d. inversely proportional to the orbiting mass.

Rodger Claar

Numerade Educator

Problem 4

Suppose you were transported to a planet with twice the mass of Earth but the same radius that Earth has. Your weight would ______ by a factor of ________.
a. increase; 2
b. increase; 4
c. decrease; 2
d. decrease; 4

Rodger Claar

Numerade Educator

Problem 5

Rank the following objects, in order of their circular velocities, from smallest to largest.
a. a $5-\mathrm{kg}$ object orbiting Earth halfway to the Moon
b. a $10-\mathrm{kg}$ object orbiting Earth just above Earth's surface
c. a 15 -kg object orbiting Earth at the same distance as the Moon
d. a 20 -kg object orbiting Earth one-quarter of the way to the Moon

Sarah Mccrumb

Numerade Educator

Problem 6

Rank the following types of orbits in terms of the maximum speed of the orbiting object, from smallest to largest.
a. elliptical, with semimajor axis $R$
b. hyperbolic
c. circular, with radius $R$
d. parabolic

Ajay Singhal

Numerade Educator

Problem 7

Which of the following objects would escape from Earth's gravity?
a. a tennis ball traveling at $41,000 \mathrm{km} / \mathrm{h}$, straight up
b. a bear traveling at $41,000 \mathrm{km} / \mathrm{h}$, straight up
c. a car traveling at $41,000 \mathrm{km} / \mathrm{h}$, straight up
d. an airplane traveling at $41,000 \mathrm{km} / \mathrm{h}$, straight up

Rodger Claar

Numerade Educator

Problem 8

Once you know the semimajor axis and the period of an orbit, you can determine
a. the mass of the orbiting object.
b. the speed of the orbiting object.
c. the mass of the object being orbited.
d. both b and c

Rodger Claar

Numerade Educator

Problem 9

Spring tides occur
a. in March, April, or May.
b. when the Moon's phase is new or full.
c. when the Moon's phase is first quarter or third quarter.
d. in either spring or fall.

Rodger Claar

Numerade Educator

Problem 10

If an object crosses the Roche limit, it
a. can no longer be seen.
b. begins to accelerate very quickly.
c. slows down.
d. may be torn apart.

Rodger Claar

Numerade Educator

Problem 11

$\mathbf{T} / \mathbf{F}:$ When an object falls to Earth, Earth also falls up to the object.

Rodger Claar

Numerade Educator

Problem 12

T/F: If a particle existed all alone in the universe, it could exert a gravitational force.

Rodger Claar

Numerade Educator

Problem 13

T/F: A satellite orbiting faster than the circular velocity will spiral in.

Rodger Claar

Numerade Educator

Problem 14

$\mathbf{T} / \mathbf{F}:$ The acceleration due to gravity is the same throughout the interior of Earth.

Rodger Claar

Numerade Educator

Problem 15

T/F: Earth has two tides each day because one is caused by the Moon and the other is caused by the Sun.

Rodger Claar

Numerade Educator

Problem 16

Venus has about 80 percent of Earth's mass and about 95 percent of Earth's radius. Your weight on Venus will be
a. 20 percent more than on Earth.
b. 20 percent less than on Earth.
c. 10 percent more than on Earth.
d. 10 percent less than on Earth.

Sarah Mccrumb

Numerade Educator

Problem 17

The escape velocity is approximately _______ as large as the circular velocity.
a. 0.4 times
b. 1.4 times
c. 14 times
d. 140 times

Ajay Singhal

Numerade Educator

Problem 18

If the Moon had twice the mass that it does, how would the strength of lunar tides change?
a. The highs would be higher, and the lows would be lower.
b. Both the highs and the lows would be higher.
c. The highs would be lower, and the lows would be higher.
d. Nothing would change.

Sarah Mccrumb

Numerade Educator

Problem 19

If Earth had half of its current radius, how would the strength of lunar tides change?
a. The highs would be higher, and the lows would be lower.
b. Both the highs and the lows would be higher.
c. The highs would be lower, and the lows would be higher.
d. Nothing would change.

Rodger Claar

Numerade Educator

Problem 20

If the Moon were 2 times closer to Earth than it is now, the gravitational force between Earth and the Moon would be
a. 2 times stronger.
b. 4 times stronger.
a. 2 times stronger.
b. 4 times stronger.

Rodger Claar

Numerade Educator

Problem 21

If the Moon were 2 times closer to Earth than it is now, the tides would be
a. 2 times stronger.
b. 4 times stronger.
c. 8 times stronger.
d. 16 times stronger.

Sarah Mccrumb

Numerade Educator

Problem 22

If two objects are tidally locked to each other,
a. the tides always stay on the same place on each object.
b. the objects always remain in the same place in each other's sky.
c. the objects are falling together.
d. both a and b

Sarah Mccrumb

Numerade Educator

Problem 23

The strongest tides occur during ________ of the Moon.
a. only the full phase
b. only the new phase
c. the full and new phases
d. the first quarter and third quarter phases

Rodger Claar

Numerade Educator

Problem 24

Self-gravity is
a. the gravitational pull of a person.
b. the force that holds objects like people and lamps together.
c. the gravitational interaction of all the parts of a body.
d. the force that holds objects on Earth.

Sarah Mccrumb

Numerade Educator

Problem 25

An object in a(n) ______ orbit in the Solar System will remain in its orbit forever. An object in a(n) ______ orbit will escape from the Solar System.
a. unbound; bound
b. circular; elliptical
c. bound; unbound
d. elliptical; circular

Sarah Mccrumb

Numerade Educator

Problem 26

Both Kepler's laws and Newton's laws tell us something about the motion of the planets, but there is a fundamental difference between them. What is that difference?

Sarah Mccrumb

Numerade Educator

Problem 27

An astronaut standing on Earth could easily lift a wrench having a mass of $1 \mathrm{kg}$, but not a scientific instrument with a mass of $100 \mathrm{kg}$. In the International Space Station, she is quite capable of manipulating both, although the scientific instrument responds much more slowly than the wrench. Explain why the scientific instrument responds more slowly.

Rodger Claar

Numerade Educator

Problem 28

Explain the difference between weight and mass.

Sarah Mccrumb

Numerade Educator

Problem 29

Weight on Earth is proportional to mass. On the Moon, too, weight is proportional to mass, but the constant of proportionality is different on the Moon than it is on Earth. Why?

Rodger Claar

Numerade Educator

Problem 30

Two comets are leaving the vicinity of the Sun, one traveling in an elliptical orbit and the other in a hyperbolic orbit. What can you say about the future of these two comets? Would you expect either of them to return eventually?

Sarah Mccrumb

Numerade Educator

Problem 31

What is the advantage of launching satellites from spaceports located near the equator? Would you expect satellites to be launched to the east or to the west? Why?

Sarah Mccrumb

Numerade Educator

Problem 32

We speak of Earth and the other planets all orbiting about the Sun. Under what circumstances do we have to consider bodies orbiting about a "common center of mass"?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 33

If you could live in a house at the center of Earth, you could float from room to room as though you were living in the International Space Station. Explain why this statement is true.

Rodger Claar

Numerade Educator

Problem 34

What determines the strength of gravity at various radii between Earth's center and its surface?

Sarah Mccrumb

Numerade Educator

Problem 35

The best time to dig for clams along the seashore is when the ocean tide is at its lowest. What phases of the Moon and times of day would be best for clam digging?

Ajay Singhal

Numerade Educator

Problem 36

The Moon is on the meridian at your seaside home, but your tide calendar does not show that it is high tide. What might explain this apparent discrepancy?

Sarah Mccrumb

Numerade Educator

Problem 37

We may have an intuitive feeling for why lunar tides raise sea level on the side of Earth facing the Moon, but why is sea level also raised on the side facing away from the Moon?

Sarah Mccrumb

Numerade Educator

Problem 38

Tides raise and lower the level of Earth's oceans. Can they do the same for Earth's landmasses? Explain your answer.

Sarah Mccrumb

Numerade Educator

Problem 39

Lunar tides raise the ocean surface less than 1 meter. How can tides as large as $5-10$ meters occur?

Sarah Mccrumb

Numerade Educator

Problem 40

Most commercial satellites are well inside the Roche limit as they orbit Earth. Why are they not torn apart?

Sarah Mccrumb

Numerade Educator

Problem 41

A typical house cat has a mass of roughly 5 kg. What is the weight of a typical house cat in newtons?

Rodger Claar

Numerade Educator

Problem 42

Mars has about one-tenth the mass of Earth, and about half of Earth's radius. What is the value of gravitational acceleration on the surface of Mars compared to that on Earth? Estimate your mass and weight on Mars compared with your mass and weight on Earth. Do Hollywood movies showing people on Mars accurately portray this change in weight?

Rodger Claar

Numerade Educator

Problem 43

Earth speeds along at $29.8 \mathrm{km} / \mathrm{s}$ in its orbit. Neptune's nearly circular orbit has a radius of $4.5 \times 10^{9} \mathrm{km},$ and the planet takes 164.8 years to make one trip around the Sun. Calculate the speed at which Neptune plods along in its orbit.

Sarah Mccrumb

Numerade Educator

Problem 44

An Earth-like planet is orbiting the bright star Vega at a distance of 1 astronomical unit (AU). The mass of Vega is twice that of the Sun.
a. How fast does the planet travel in its orbit?
b. What is the period of the planet?

Sarah Mccrumb

Numerade Educator

Problem 45

Venus's circular velocity is $35.03 \mathrm{km} / \mathrm{s}$, and its orbital radius is $1.082 \times 10^{8} \mathrm{km} .$ Calculate the mass of the Sun.

Sarah Mccrumb

Numerade Educator

Problem 46

At the surface of Earth, the escape velocity is $11.2 \mathrm{km} / \mathrm{s}$ What would be the escape velocity at the surface of a very small asteroid having a radius $10^{-4}$ that of Earth's and a mass $10^{-12}$ that of Earth's?

Sarah Mccrumb

Numerade Educator

Problem 47

How long does it take Newton's cannonball, moving at $7.9 \mathrm{km} / \mathrm{s}$ just above Earth's surface, to complete one orbit around Earth?

Sarah Mccrumb

Numerade Educator

Problem 48

When a spacecraft is sent to Mars, it is first launched into an Earth orbit with circular velocity.
a. Describe the shape of this orbit.
b. What minimum velocity must we give the spacecraft to send it on its way to Mars?

Sarah Mccrumb

Numerade Educator

Problem 49

Earth's mean radius and mass are $6,370 \mathrm{km}$ and $5.97 \times 10^{24} \mathrm{kg}$ respectively. Show that the acceleration of gravity at the surface of Earth is $9.81 \mathrm{m} / \mathrm{s}^{2}$

Sarah Mccrumb

Numerade Educator

Problem 50

Using $6,370 \mathrm{km}$ for Earth's radius, compare the gravitational force acting on a NASA rocket when it's sitting on its launchpad with the gravitational force acting on it when it is orbiting $350 \mathrm{km}$ above Earth's surface.

Sarah Mccrumb

Numerade Educator

Problem 51

The International Space Station travels on a nearly circular orbit $350 \mathrm{km}$ above Earth's surface. What is its orbital speed?

Sarah Mccrumb

Numerade Educator

Problem 52

As described in Math Tools $4.4,$ tidal force is proportional to the masses of the two objects and is inversely proportional to the cube of the distance between them. Some astrologers claim that your destiny is determined by the "influence" of the planets that are rising above the horizon at the moment of your birth. Compare the tidal force of Jupiter (mass $1.9 \times 10^{27} \mathrm{kg}$; distance $7.8 \times 10^{11}$ meters with that of the doctor in attendance at your birth (mass $80 \mathrm{kg}, \text { distance } 1 \text { meter })$

Sarah Mccrumb

Numerade Educator

Problem 53

The asteroid Ida (mass $4.2 \times 10^{16} \mathrm{kg}$ ) is attended by a tiny asteroidal moon, Dactyl, which orbits Ida at an average distance of $90 \mathrm{km}$. Neglecting the mass of the tiny moon, what is Dactyl's orbital period in hours?

Sarah Mccrumb

Numerade Educator

Problem 54

The two stars in a binary star system (two stars orbiting about a common center of mass) are separated by $10^{\circ} \mathrm{km}$ and revolve around their common center of mass in 10 years.
a. What is their combined mass, in kilograms?
b. The mass of the Sun is $1.99 \times 10^{30} \mathrm{kg}$. What is the combined mass of the two stars, in solar masses?

Julie Farhm

Numerade Educator

Problem 55

Suppose you go skydiving.
a. Just as you fall out of the airplane, what is your gravitational acceleration?
b. Would this acceleration be bigger, smaller, or the same if you were strapped to a flight instructor, and so had twice the mass?
c. Just as you fall out of the airplane, what is the gravitational force on you? (Assume your mass is 70 kg.)
d. Would the gravitational force be bigger, smaller, or the same if you were strapped to a flight instructor, and so had twice the mass?

Sarah Mccrumb

Numerade Educator

Problem 56

Go to NASA's "Apollo 15 Hammer-Feather Drop" Web page (http://nssdc.gsfc.nasa.gov/planetary/lunar/apollo_15_ feather_drop.html) and watch the video from Apollo 15 of astronaut David Scott dropping the hammer and falcon feather on the Moon. (You might find a better version on YouTube.) What did this experiment show? What would happen if you tried this on Earth with a feather and a hammer? Would it work? What would you see? Suppose instead you dropped the hammer and a big nail. How would they fall? How fast do things fall on the Moon compared to on Earth?

Rodger Claar

Numerade Educator

Problem 57

Go to the Exploratorium's "Your Weight on Other Worlds" page (http://exploratorium.edu/ronh/weight), which will calculate your weight on other planets and moons in our Solar System. On which objects would your weight be higher than it is on Earth? What difficulties would human bodies have in a higher-gravity environment? For example, would it be easy to get up out of bed and walk? What are the possible short-term and long-term effects of lower gravity on the human body? Can you think of some types of life on Earth that might adapt well to a different gravity?

Rodger Claar

Numerade Educator

Problem 58

Go to a website that will show you the times for high and low tides in your area-for example, http://saltwatertides com. Pick a location and bring up the tide table for today and the next 2 weeks. Why are there two high tides and two low tides every day? What is the difference in the height of the water between high and low tides?

Rodger Claar

Numerade Educator

Problem 59

Go to a website with the phases of the Moon and the times of moonrise and moonset for your location-for example,
http://aa.usno.navy.mil/data/docs/RS_OneYear.php. Does the time of the high tide lead or follow the position of the Moon?

Chris Johnson

Numerade Educator