College Physics: A Strategic Approach

Randall D. Knight, Brian Jones, Stuart Field

Chapter 6

Circular Motion, Orbits, and Gravity - all with Video Answers

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Chapter Questions

Problem 1

A 5.0-m-diameter merry-go-round is turning with a 4.0 s period. What is the speed of a child on the rim?

Vishal Gupta

Vishal Gupta

Numerade Educator

Problem 2

The earth, with a radius of $6.4 \times 10^{6} \mathrm{m},$ rotates on its axis once a day. What is the speed of a person standing on the equator, due to the earth's rotation?

Suhas Katkar

Numerade Educator

Problem 3

An old-fashioned LP record rotates at $33 \frac{1}{3}$ rpm.
a. What is its frequency, in rev/s?
b. What is its period, in seconds?

Suhas Katkar

Numerade Educator

Problem 4

A typical hard disk in a computer spins at 5400 rpm.
a. What is the frequency, in rev/s?
b. What is the period, in seconds?

Suhas Katkar

Numerade Educator

Problem 5

The hammer throw was one of the earliest Olympic events. In this event, a heavy ball attached to a chain is swung several times in a circular path until it is released. The winning athlete is the one who throws the ball the greatest distance. The last complete rotation of 2016 Olympic champion Anita Whodarczyk's final turn took only 0.43 s. The radius of the ball's path, including her extended arms, was $2.1 \mathrm{m}$.
a. What was the frequency of this rotation?
b. What was the speed of the ball?
c. What was the ball's acceleration, in units of g ?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 6

The horse on a carousel is $4.0 \mathrm{m}$ from the central axis.
a. If the carousel rotates at 0.10 rev/s, how long does it take the horse to go around twice?
b. How fast is a child on the horse going (in $\mathrm{m} / \mathrm{s}$ )?

Suhas Katkar

Numerade Educator

Problem 7

The radius of the earth's very nearly circular orbit around the sun is $1.50 \times 10^{11} \mathrm{m} .$ Find the magnitude of the earth's (a) velocity and (b) centripetal acceleration as it travels around the sun. Assume a year of 365 days.

Suhas Katkar

Numerade Educator

Problem 8

Modern wind turbines are larger than they appear, and despite their apparently lazy motion, the speed of the blades tips can be quite high-many times higher than the wind speed. A typical modern turbine has blades $56 \mathrm{m}$ long that spin at 13 rpm. At the tip of a blade, what are (a) the speed and (b) the centripetal acceleration?

Anand Jangid

Numerade Educator

Problem 9

The California sea lion is capable of making extremely fast, tight turns while swimming underwater. In one study, scientists observed a sea lion making a circular turn with a radius of $0.35 \mathrm{m}$ while swimming at $4.2 \mathrm{m} / \mathrm{s}$.
a. What is the sea lion's centripetal acceleration, in units of $g ?$
b. What percentage is this acceleration of that of an $\mathrm{F}-15$ fighter jet's maximum centripetal acceleration of $9 g ?$

Suhas Katkar

Numerade Educator

Problem 10

Baseball pitching machines are used to fire baseballs toward a batter for hitting practice. In one kind of machine, the ball is fed between two wheels that are rapidly rotating in opposite directions; as the ball is pulled in between the wheels, it rapidly accelerates up to the speed of the wheels' rims, at which point it is ejected. For a machine with 35 -cm-diameter wheels, what rotational frequency (in rpm) do the wheels need to pitch a 90 mph fastball?

Suhas Katkar

Numerade Educator

Problem 11

Astronauts in the International Space Station must work out every day to counteract the effects of weightlessness. Researchers have investigated if riding a stationary bicycle while experiencing artificial gravity from a rotating platform gives any additional cardiovascular benefit. What frequency of rotation, in rpm, is required to give an acceleration of $1.4 g$ to an astronaut's feet, if her feet are $1.1 \mathrm{m}$ from the platform's rotational axis?

Suhas Katkar

Numerade Educator

Problem 12

A typical running track is an oval with 74 -m-diameter half circles at each end. A runner going once around the track covers a distance of $400 \mathrm{m} .$ Suppose a runner, moving at a constant speed, goes once around the track in 1 min 40 s. What is her centripetal acceleration during the turn at each end of the track?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 13

Figure $\mathrm{P} 6.13$ is a bird's-eye view of particles on a string moving in horizontal circles on a tabletop. All are moving at the same speed. Rank in order, from largest to smallest, the tensions $T_{1}$ to $T_{4}$.

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 14

In short-track speed skating, the track has straight sections and semicircles $16 \mathrm{m}$ in diameter. Assume that a $65 \mathrm{kg}$ skater goes around the turn at a constant $12 \mathrm{m} / \mathrm{s}$.
a. What is the horizontal force on the skater?
b. What is the ratio of this force to the skater's weight?

Suhas Katkar

Numerade Educator

Problem 15

In addition to their remarkable top speeds of almost $60 \mathrm{mph}$, cheetahs have impressive cornering abilities. In one study, the maximum centripetal acceleration of a cheetah was measured to be $18 \mathrm{m} / \mathrm{s}^{2} .$ What minimum value of the coefficient of static friction between the ground and the cheetah's feet is necessary to provide this acceleration?

Supratim Pal

Numerade Educator

Problem 16

A cyclist is rounding a 20 -m-radius curve at $12 \mathrm{m} / \mathrm{s}$. What is the minimum possible coefficient of static friction between the bike tires and the ground?

Suhas Katkar

Numerade Educator

Problem 17

$\mathrm{A} 1500 \mathrm{kg}$ car drives around a flat $200-\mathrm{m}$ -diameter circular track at $25 \mathrm{m} / \mathrm{s}$. What are the magnitude and direction of the net force on the car? What causes this force?

Suhas Katkar

Numerade Educator

Problem 18

A fast pitch softball player does a "windmill" pitch, illustrated in Figure $\mathrm{P} 6.18$, moving her hand through a circular arc to pitch a ball at 70 mph. The 0.19 kg ball is 50 cm from the pivot point at her shoulder. At the lowest point of the circle, the ball has reached its maximum speed.
a. At the bottom of the circle, just before the ball leaves her hand, what is its centripetal acceleration?
b. What are the magnitude and direction of the force her hand exerts on the ball at this point?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 19

One kind of baseball pitching machine works by rotating a light and stiff rigid rod about a horizontal axis until the ball is moving toward the target. Suppose a 144 g baseball is held 85 $\mathrm{cm}$ from the axis of rotation and released at the major league pitching speed of 85 mph.
a. What is the ball's centripetal acceleration just before it is released?
b. What is the magnitude of the net force that is acting on the ball just before it is released?

Suhas Katkar

Numerade Educator

Problem 20

A wind turbine has 12,000 kg blades that are 38 m long. The blades spin at 22 rpm. If we model a blade as a point mass at the midpoint of the blade, what is the inward force necessary to provide each blade's centripetal acceleration?

Suhas Katkar

Numerade Educator

Problem 21

You're driving your pickup truck around a curve that has a radius of $20 \mathrm{m}$. How fast can you drive around this curve before a steel toolbox slides on the steel bed of the truck?

Suhas Katkar

Numerade Educator

Problem 22

The spin cycle of a clothes washer extracts the water in clothing by greatly increasing the water's apparent weight so that it is efficiently squeezed through the clothes and out the holes in the drum. In a top loader's spin cycle, the 45-cm-diameter drum spins at 1200 rpm around a vertical axis. What is the apparent weight of a $1.0 \mathrm{g}$ drop of water?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 23

Gibbons, small Asian apes, move by brachiation, swinging below a handhold to move forward to the next handhold. A 9.0 kg gibbon has an arm length (hand to shoulder) of $0.60 \mathrm{m}$. We can model its motion as that of a point mass swinging at the end of a $0.60-\mathrm{m}$ -long, massless rod. At the lowest point of its swing, the gibbon is moving at $3.5 \mathrm{m} / \mathrm{s}$. What upward force must a branch provide to support the swinging gibbon?

Suhas Katkar

Numerade Educator

Problem 24

The passengers in a roller coaster car feel $50 \%$ heavier than their true weight as the car goes through a dip with a 30 m radius of curvature. What is the car's speed at the bottom of the dip?

Suhas Katkar

Numerade Educator

Problem 25

In the very Dutch sport of Fierljeppen, athletes run up to a long pole and then use it to vault across a canal. At the very top of his arc, a $55 \mathrm{kg}$ vaulter is moving at $2.5 \mathrm{m} / \mathrm{s}$ and is $5.1 \mathrm{m}$ from the bottom end of the pole. What vertical force does the pole exert on the vaulter?

Suhas Katkar

Numerade Educator

Problem 26

A roller coaster car is going over the top of a 15 -m-radius circular rise. At the top of the hill, the passengers "feel light," with an apparent weight only $50 \%$ of their true weight. How fast is the coaster moving?

Suhas Katkar

Numerade Educator

Problem 27

As a roller coaster car crosses the top of a 40-m-diameter loop-the-loop, its apparent weight is the same as its true weight. What is the car's speed at the top?

Suhas Katkar

Numerade Educator

Problem 28

Unlike a roller coaster, the seats in a Ferris wheel swivel so that the rider is always seated upright. An 80 -ft-diameter Ferris wheel rotates once every 24 s. What is the apparent weight of a $70 \mathrm{kg}$ passenger at $(\mathrm{a})$ the lowest point of the circle and $(\mathrm{b})$ the highest point?

Suhas Katkar

Numerade Educator

Problem 29

You're driving your new sports car at 75 mph over the top of a hill that has a radius of curvature of $525 \mathrm{m} .$ What fraction of your normal weight is your apparent weight as you crest the hill?

Narayan Hari

Numerade Educator

Problem 30

In a car's suspension system, each wheel is connected to a vertical spring; these springs absorb shocks when the car travels on bumpy roads. In one car, each spring has a spring constant of $5.0 \times 10^{4} \mathrm{N} / \mathrm{m} .$ If this $1400 \mathrm{kg}$ car is driven at $25 \mathrm{m} / \mathrm{s}$ through the bottom of a circular dip in the road that has a radius of $600 \mathrm{m},$ by how much do these springs compress compared to when the car is driven on a flat road?

Suhas Katkar

Numerade Educator

Problem 31

A typical laboratory centrifuge rotates at 4000 rpm. Test tubes have to be placed into a centrifuge very carefully because of the very large accelerations.
a. What is the acceleration at the end of a test tube that is $10 \mathrm{cm}$ from the axis of rotation?
b. For comparison, what is the magnitude of the acceleration a test tube would experience if stopped in a 1.0 -ms-long encounter with a hard floor after falling from a height of $1.0 \mathrm{m} ?$

Suhas Katkar

Numerade Educator

Problem 32

A satellite orbiting the moon very near the surface has a period of 110 min. Use this information, together with the radius of the moon from the table on the inside of the back cover, to calculate the free-fall acceleration on the moon's surface.

Suhas Katkar

Numerade Educator

Problem 33

Many spacecraft have visited Mars over the years. Mars is smaller than Earth and has correspondingly weaker surface gravity. On Mars, the free-fall acceleration is only $3.8 \mathrm{m} / \mathrm{s}^{2}$. What is the orbital period of a spacecraft in a low orbit near the surface of Mars?

Suhas Katkar

Numerade Educator

Problem 34

The centers of a $10 \mathrm{kg}$ lead ball and a $100 \mathrm{g}$ lead ball are separated by $10 \mathrm{cm} .$
a. What gravitational force does each exert on the other?
b. What is the ratio of this gravitational force to the weight of the 100 g ball?

Suhas Katkar

Numerade Educator

Problem 35

The gravitational force of a star on an orbiting planet 1 is $F_{1} .$ Planet $2,$ which is twice as massive as planet 1 and orbits at twice the distance from the star, experiences gravitational force $F_{2} .$ What is the ratio $F_{2} / F_{1} ?$ You can ignore the gravitational force between the two planets.

Suhas Katkar

Numerade Educator

Problem 36

The free-fall acceleration at the surface of planet 1 is $20 \mathrm{m} / \mathrm{s}^{2}$. The radius and the mass of planet 2 are twice those of planet $1 .$ What is the free-fall acceleration on planet $2 ?$

Suhas Katkar

Numerade Educator

Problem 37

What is the ratio of the sun's gravitational force on you to the earth's gravitational force on you?

Suhas Katkar

Numerade Educator

Problem 38

Just before it landed on the moon, the Apollo 12 lunar lander had a mass of $7200 \mathrm{kg}$. What rocket thrust was necessary to have the lander touch down with zero acceleration?

Suhas Katkar

Numerade Educator

Problem 39

In recent years, astronomers have found planets orbiting nearby stars that are quite different from planets in our solar system. Kepler-12b, has a diameter that is 1.7 times that of Jupiter, but a mass that is only 0.43 that of Jupiter. What is the value of $g$ on this large, but low-density, world?

Suhas Katkar

Numerade Educator

Problem 40

The gravitational constant $G$ was first measured accurately by Henry Cavendish in $1798 .$ He used an exquisitely sensitive balance to measure the force between two lead spheres whose centers were $0.23 \mathrm{m}$ apart. One of the spheres had a mass of $158 \mathrm{kg},$ while the mass of the other sphere was $0.73 \mathrm{kg} .$ What was the ratio of the gravitational force between these spheres to the weight of the lighter sphere?

Suhas Katkar

Numerade Educator

Problem 41

a. What is the gravitational force of the sun on the earth?
b. What is the gravitational force of the moon on the earth?
c. The moon's force is what percent of the sun's force?

Suhas Katkar

Numerade Educator

Problem 42

What is the value of $g$ on the surface of Saturn? Explain how such a low value is possible given Saturn's large mass-100 times that of Earth.

Suhas Katkar

Numerade Educator

Problem 43

What is the free-fall acceleration at the surface of (a) Mars and (b) Jupiter?

Suhas Katkar

Numerade Educator

Problem 44

In $2014,$ a space probe approached the rocky core of the comet Churyumov-Gerasimenko, which is only a few $\mathrm{km}$ in diameter. The probe then entered orbit around the comet at a distance of $30 \mathrm{km} .$ The comet was found to have a mass of $1.0 \times 10^{13} \mathrm{kg} .$ What was the orbital period of the probe around the comet, in earth days?

Suhas Katkar

Numerade Educator

Problem 45

Mars has two moons, Phobos and Deimos. Phobos orbits Mars at a distance of $9380 \mathrm{km}$ from Mars's center, while Deimos orbits at $23,500 \mathrm{km}$ from the center. What is the ratio of the orbital period of Deimos to that of Phobos?

Suhas Katkar

Numerade Educator

Problem 46

Mars's moon Phobos orbits the planet at a distance of 9380 $\mathrm{km}$ from its center, and it takes 7 hours and 39 minutes to complete one orbit. What is the ratio of Mars's mass to the mass of the earth?

Suhas Katkar

Numerade Educator

Problem 47

The dwarf planet Praamzius is estimated to have a diameter of about $300 \mathrm{km}$ and orbits the sun at a distance of $6.4 \times 10^{12} \mathrm{m}$ What is its orbital period in years?

Suhas Katkar

Numerade Educator

Problem 48

Planet $X$ orbits the star Omega with a "year" that is 200 earth days long. Planet $Y$ circles Omega at four times the distance of Planet X. How long is a year on Planet Y?

Suhas Katkar

Numerade Educator

Problem 49

The International Space Station is in a 250 -mile-high orbit. What are the station's orbital period, in minutes, and speed?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 50

An earth satellite moves in a circular orbit at a speed of $5500 \mathrm{m} / \mathrm{s} .$ What is its orbital period?

Suhas Katkar

Numerade Educator

Problem 51

In recent years, scientists have discovered hundreds of planets orbiting other stars. Some of these planets are in orbits that are similar to that of earth, which orbits the sun $\left(M_{\text {sun }}=1.99 \times 10^{30} \mathrm{kg}\right)$ at a distance of $1.50 \times 10^{11} \mathrm{m},$ called 1 astronomical unit $(1$ au $) .$ Others have extreme orbits that are much different from anything in our solar system. Problems $51-53$ relate to some of these planets that follow circular orbits around other stars.
WASP-32b orbits with a period of only 2.7 days a star with a mass that is 1.1 times that of the sun. How many au from the star is this planet?

Suhas Katkar

Numerade Educator

Problem 52

In recent years, scientists have discovered hundreds of planets orbiting other stars. Some of these planets are in orbits that are similar to that of earth, which orbits the sun $\left(M_{\text {sun }}=1.99 \times 10^{30} \mathrm{kg}\right)$ at a distance of $1.50 \times 10^{11} \mathrm{m},$ called 1 astronomical unit $(1$ au $) .$ Others have extreme orbits that are much different from anything in our solar system. Problems $51-53$ relate to some of these planets that follow circular orbits around other stars.
HD $10180 \mathrm{g}$ orbits with a period of 600 days at a distance of 1.4 au from its star. What is the ratio of the star's mass to our sun's mass?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 53

In recent years, scientists have discovered hundreds of planets orbiting other stars. Some of these planets are in orbits that are similar to that of earth, which orbits the sun $\left(M_{\text {sun }}=1.99 \times 10^{30} \mathrm{kg}\right)$ at a distance of $1.50 \times 10^{11} \mathrm{m},$ called 1 astronomical unit $(1$ au $) .$ Others have extreme orbits that are much different from anything in our solar system. Problems $51-53$ relate to some of these planets that follow circular orbits around other stars.
Kepler-42c orbits at a very close 0.0058 au from a small star with a mass that is 0.13 that of the sun. How long is a "year" on this world?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 54

How fast must a plane fly along the earth's equator so that the sun stands still relative to the passengers? In which direction must the plane fly, east to west or west to east? Give your answer in both $\mathrm{km} / \mathrm{h}$ and $\mathrm{mph} .$ The radius of the earth is $6400 \mathrm{km}$.

Suhas Katkar

Numerade Educator

Problem 55

The car in Figure $\mathrm{P} 6.55$ travels at a constant speed along the road shown. Draw vectors showing its acceleration at the three points $\mathrm{A}, \mathrm{B},$ and $\mathrm{C},$ or write $\vec{a}=\overrightarrow{0} .$ The lengths of your vectors should correspond to the magnitudes of the accelerations.

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 56

$\mathrm{A}$ 75 $\mathrm{kg}$ man weighs himself at the north pole and at the equator. Which scale reading is higher? By how much? Assume the earth is a perfect sphere. Explain why the readings differ.

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 57

$\mathrm{A} 1400 \mathrm{kg}$ car drives at $27 \mathrm{m} / \mathrm{s}$ over a circular hill that has a a a cular hill that has a radius of $430 \mathrm{m}$. At the point shown in Figure $\mathrm{P} 6.57,$ what is the normal force on the car?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 58

A curve at a racetrack has a radius of $800 \mathrm{m}$ and is banked at an angle of $7.0^{\circ} .$ On a rainy day, the coefficient of friction between the cars' tires and the track is 0.50 . What is the maximum speed at which a car could go around this curve without slipping?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 59

It is well known that runners run more slowly around a curved track than a straight one. One hypothesis to explain this is that the total force from the track on a runner's feet-the magnitude of the vector sum of the normal force (that has average value $m g$ to counteract gravity) and the inward-directed friction force that causes the runner's centripetal accelerationis greater when running around a curve than on a straight track. Runners compensate for this greater force by increasing the time their feet are in contact with the ground, which slows them down. For a sprinter running at $10 \mathrm{m} / \mathrm{s}$ around a curved track of radius $20 \mathrm{m},$ how much greater (as a percentage) is the average total force on their feet compared to when they are running in a straight line?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 60

Biologists have studied the running ability of the northern quoll, a marsupial indigenous to Australia. In one set of experiments, they studied the maximum speed that quolls could run around a curved path without slipping. One quoll was running at $2.8 \mathrm{m} / \mathrm{s}$ around a curve with a radius of $1.2 \mathrm{m}$ when it started to slip. What was the coefficient of static friction between the quoll's feet and the ground in this trial?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 61

In a recent study of how mice negotiate turns, the mice ran around a circular $90^{\circ}$ turn on a track with a radius of $0.15 \mathrm{m} .$ The maximum speed measured for a mouse (mass $=18.5$ g $)$ running around this turn was $1.29 \mathrm{m} / \mathrm{s}$. What is the minimum coefficient of friction between the track and the mouse's feet that would allow a turn at this speed?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 62

You are driving your car through a roundabout that has a radius of $9.0 \mathrm{m} .$ Your physics textbook is lying on the seat next to you. What is the fastest speed at which you can go around the curve without the book sliding? The coefficient of static friction between the book and the seat is 0.30 .

Suhas Katkar

Numerade Educator

Problem 63

In the Wall of Death carnival attraction, stunt motorcyclists ride around the inside of a large, 10 -m-diameter wooden cylinder that has vertical walls. The coefficient of static friction between the riders' tires and the wall is $0.90 .$ What is the minimum speed at which the motorcyclists can ride without slipping down the wall?

Suhas Katkar

Numerade Educator

Problem 64

In the swing carousel park ride, riders sit in chairs that are attached by a chain to a large rotating drum. As the carousel turns, the riders move in a large circle with the chains tilted out from the vertical. In one such carousel, the riders move in a 16.5-m-radius circle and take 8.3 s to complete one revolution. What is the angle of the chains, as measured from the vertical?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 65

$\mathrm{A} 5.0 \mathrm{g}$ coin is placed $15 \mathrm{cm}$ from the center of a turntable. The coin has static and kinetic coefficients of friction with the turntable surface of $\mu_{\mathrm{s}}=0.80$ and $\mu_{\mathrm{k}}=0.50 .$ The turntable very slowly speeds up to 60 rpm. Does the coin slide off?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 66

In an old-fashioned amusement park ride, passengers stand inside a $3.0-\mathrm{m}$ -tall, 5.0 - $\mathrm{m}$ -diameter hollow steel cylinder with their backs against the wall. The cylinder begins to rotate about a vertical axis. Then the floor on which the passengers are standing suddenly drops away! If all goes well, the passengers will "stick" to the wall and not slide. Clothing has a static coefficient of friction against steel in the range 0.60 to 1.0 and a kinetic coefficient in the range 0.40 to $0.70 .$ What is the minimum rotational frequency, in rpm, for which the ride is safe?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 67

The $0.20 \mathrm{kg}$ puck on the frictionless, horizontal table in Figure $\mathrm{P} 6.67$ is connected by a string through a hole in the table to a hanging $1.20 \mathrm{kg}$ block. With what speed must the puck rotate in a circle of radius 0.50 $\mathrm{m}$ if the block is to remain hanging at rest?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 68

While at the county fair, you decide to ride the Ferris wheel. Having eaten too many candy apples and elephant ears, you find the motion somewhat unpleasant. To take your mind off your stomach, you wonder about the motion of the ride. You estimate the radius of the big wheel to be $15 \mathrm{m},$ and you use your watch to find that each loop around takes 25 s.
a. What are your speed and magnitude of your acceleration?
b. What is the ratio of your apparent weight to your true weight at the top of the ride?
c. What is the ratio of your apparent weight to your true weight at the bottom?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 69

A car drives over the top of a hill that has a radius of $50 \mathrm{m}$. What maximum speed can the car have without flying off the road at the top of the hill?

Suhas Katkar

Numerade Educator

Problem 70

The ultracentrifuge is an important tool for separating and analyzing proteins in biological research. Because of the enor mous centripetal accelerations that can be achieved, the apparatus (see Figure 6.18 ) must be carefully balanced so that each sample is matched by another on the opposite side of the rotor shaft. Any difference in mass of the opposing samples will cause a net force in the horizontal plane on the shaft of the rotor; this force can actually be large enough to destroy the centrifuge. Suppose that a scientist makes a slight error in sample preparation, and one sample has a mass 10 mg greater than the opposing sample. If the samples are $10 \mathrm{cm}$ from the axis of the rotor and the ultracentrifuge spins at 70,000 rpm, what is the magnitude of the net force on the rotor due to the unbalanced samples?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 71

A sensitive gravimeter at a mountain observatory finds that the free-fall acceleration is $0.0075 \mathrm{m} / \mathrm{s}^{2}$ less than that at sea level. What is the observatory's altitude?

Suhas Katkar

Numerade Educator

Problem 72

In 2014 , the Rosetta space probe reached the comet ChuryumovGerasimenko. Although the comet's core is actually far from spherical, in this problem we'll model it as a sphere with a mass of $1.0 \times 10^{13} \mathrm{kg}$ and a radius of $1.6 \mathrm{km} .$ If a rock were dropped from a height of $1.0 \mathrm{m}$ above the comet's surface, how long would it take to hit the surface?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 73

Planet $Z$ is $10,000 \mathrm{km}$ in diameter. The free-fall acceleration on Planet $\mathrm{Z}$ is $8.0 \mathrm{m} / \mathrm{s}^{2}$.
a. What is the mass of Planet Z?
b. What is the free-fall acceleration $10,000 \mathrm{km}$ above Planet $\mathrm{Z}$ 's north pole?

Suhas Katkar

Numerade Educator

Problem 74

How long will it take a rock dropped from $2.0 \mathrm{m}$ above the surface of Mars to reach the ground?

Suhas Katkar

Numerade Educator

Problem 75

A $20 \mathrm{kg}$ sphere is at the origin and a $10 \mathrm{kg}$ sphere is at $(x, y)=(20 \mathrm{cm}, 0 \mathrm{cm}) .$ At what point or points could you place a small mass such that the net gravitational force on it due to the spheres is zero?

Suhas Katkar

Numerade Educator

Problem 76

a. At what height above the earth is the free-fall acceleration $10 \%$ of its value at the surface?
b. What is the speed of a satellite orbiting at that height?

Suhas Katkar

Numerade Educator

Problem 77

Mars has a small moon, Phobos, that orbits with a period of $7 \mathrm{h} 39 \mathrm{min} .$ The radius of Phobos' orbit is $9.4 \times 10^{6} \mathrm{m} .$ Use only this information (and the value of $G$ ) to calculate the mass of Mars.

Suhas Katkar

Numerade Educator

Problem 78

You are the science officer on a visit to a distant solar system. Prior to landing on a planet you measure its diameter to be $1.80 \times 10^{7} \mathrm{m}$ and its rotation period to be $22.3 \mathrm{h} .$ You have previously determined that the planet orbits $2.20 \times 10^{11} \mathrm{m}$ from its star with a period of 402 earth days. Once on the surface you find that the free-fall acceleration is $12.2 \mathrm{m} / \mathrm{s}^{2} .$ What are the masses of (a) the planet and (b) the star?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 79

Europa, a satellite of Jupiter, is believed to have a liquid ocean of water (with a possibility of life) beneath its icy surface. In planning a future mission to Europa, what is the fastest that an astronaut with legs of length $0.70 \mathrm{m}$ could walk on the surface of Europa? Europa is $3100 \mathrm{km}$ in diameter and has a mass of $4.8 \times 10^{22} \mathrm{kg}$.

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 80

Suppose a spacecraft orbits the moon in a very low, circular orbit, just a few hundred meters above the lunar surface. The moon has a diameter of $3500 \mathrm{km},$ and the free-fall acceleration at the surface is $1.6 \mathrm{m} / \mathrm{s}^{2}$.
The direction of the net force on the craft is
A. Away from the surface of the moon.
B. In the direction of motion.
C. Toward the center of the moon.
D. Nonexistent, because the net force is zero.

Suhas Katkar

Numerade Educator

Problem 81

Suppose a spacecraft orbits the moon in a very low, circular orbit, just a few hundred meters above the lunar surface. The moon has a diameter of $3500 \mathrm{km},$ and the free-fall acceleration at the surface is $1.6 \mathrm{m} / \mathrm{s}^{2}$.
How fast is this spacecraft moving?
A. $53 \mathrm{m} / \mathrm{s}$
B. $75 \mathrm{m} / \mathrm{s}$
C. $1700 \mathrm{m} / \mathrm{s}$
D. $2400 \mathrm{m} / \mathrm{s}$

Suhas Katkar

Numerade Educator

Problem 82

Suppose a spacecraft orbits the moon in a very low, circular orbit, just a few hundred meters above the lunar surface. The moon has a diameter of $3500 \mathrm{km},$ and the free-fall acceleration at the surface is $1.6 \mathrm{m} / \mathrm{s}^{2}$.
How much time does it take for the spacecraft to complete one orbit?
A. 38 min
B. 76 min
C. 110 min
D. 220 min

Suhas Katkar

Numerade Educator

Problem 83

The material that comprises the side of the moon facing the earth is actually slightly more dense than the material on the far side. When the spacecraft is above a more dense area of the surface, the moon's gravitational force on the craft is a bit stronger. In order to stay in a circular orbit of constant height and speed, the spacecraft could fire its rockets while passing over the denser area. The rockets should be fired so as to generate a force on the craft
A. Away from the surface of the moon.
B. In the direction of motion.
C. Toward the center of the moon.
D. Opposite the direction of motion.

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator