Principles of Physics a Calculus Based Text

Raymond A. Serway, John W. Jewett, Jr.

Chapter 11

Gravity, Planetary Orbits, and the Hydrogen Atom - all with Video Answers

Educators

Chapter Questions

Problem 1

Two ocean liners, each with a mass of 40000 metric tons, are moving on parallel courses $100 \mathrm{m}$ apart. What is the magnitude of the acceleration of one of the liners toward the other due to their mutual gravitational attraction? Model the ships as particles.

Samantha Kamal

Samantha Kamal

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Problem 2

During a solar eclipse, the Moon, the Earth, and the Sun all lie on the same line, with the Moon between the Earth and the Sun. (a) What force is exerted by the Sun on the Moon? (b) What force is exerted by the Earth on the Moon? (c) What force is exerted by the Sun on the Earth? (d) Compare the answers to parts (a) and (b). Why doesn't the Sun capture the Moon away from the Earth?

Nathan Nowack

Numerade Educator

Problem 3

$ A $200 -kg object and a 500 -kg object are separated by $4.00 \mathrm{m} .$
(a) Find the net gravitational force exerted by these objects on a 50.0 -kg object placed midway between them.
(b) At what position (other than an infinitely remote one) can the 50.0 -kg object be placed so as to experience a net force of zero from the other two objects?

Nathan Nowack

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Problem 4

A satellite of mass $300 \mathrm{kg}$ is in a circular orbit around the Earth at an altitude equal to the Earth's mean radius. Find
(a) the satellite's orbital speed, (b) the period of its revolution, and (c) the gravitational force acting on it.

Nathan Nowack

Numerade Educator

Problem 5

In introductory physics laboratories, a typical Cavendish balance for measuring the gravitational constant $G$ uses lead spheres with masses of $1.50 \mathrm{kg}$ and $15.0 \mathrm{g}$ whose centers are separated by about $4.50 \mathrm{cm} .$ Calculate the gravitational force between these spheres, treating each as a particle located at the sphere's center.

Nathan Nowack

Numerade Educator

Problem 6

A student proposes to study the gravitational force by suspending two 100.0 -kg spherical objects at the lower ends of cables from the ceiling of a tall cathedral and measuring the deflection of the cables from the vertical. The 45.00 -m-long cables are attached to the ceiling $1.000 \mathrm{m}$ apart. The first object is suspended, and its position is carefully measured. The second object is suspended, and the two objects attract each other gravitationally. By what distance has the first object moved horizontally from its initial position due to the gravitational attraction to the other object? Suggestion: Keep in mind that this distance will be very small and make appropriate approximations.

Donald Albin

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Problem 7

The free-fall acceleration on the surface of the Moon is about one-sixth that on the surface of the Earth. The radius of the Moon is about $0.250 R_{E}\left(R_{E}=\text { Earth's radius }=\right.$ $6.37 \times 10^{6} \mathrm{m}$ ). Find the ratio of their average densities, $\rho_{\text {Moon }} / \rho_{\text {Earth }}$

Donald Albin

Numerade Educator

Problem 8

Why is the following situation impossible? The centers of two homogeneous spheres are $1.00 \mathrm{m}$ apart. The spheres are each made of the same element from the periodic table. The gravitational force between the spheres is $1.00 \mathrm{N}$.

Donald Albin

Numerade Educator

Problem 9

Miranda, a satellite of Uranus, is shown in Figure $\mathrm{P} 11.9 \mathrm{a}$. It can be modeled as a sphere of radius $242 \mathrm{km}$ and mass $6.68 \times 10^{19} \mathrm{kg}$. (a) Find the free-fall acceleration on its surface. (b) A cliff on Miranda is $5.00 \mathrm{km}$ high. It appears on the limb at the 11 o'clock position in Figure $\mathrm{P} 11.9 \mathrm{a}$ and is magnified in Figure $\mathrm{P} 11.9 \mathrm{b}$. If a devotee of extreme sports runs horizontally off the top of the cliff at $8.50 \mathrm{m} / \mathrm{s}$ for what time interval is he in flight? (c) How far from the base of the vertical cliff does he strike the icy surface of Miranda? (d) What will be his vector impact velocity?

Nathan Nowack

Numerade Educator

Problem 10

Three uniform spheres of masses $m_{1}=2.00 \mathrm{kg}, m_{2}=$ $4.00 \mathrm{kg},$ and $m_{3}=6.00 \mathrm{kg}$ are placed at the corners of a right triangle as shown in Figure P11.10. Calculate the resultant gravitational force on the object of mass $m_{2},$ assuming the spheres are isolated from the rest of the Universe.

Donald Albin

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Problem 11

A spacecraft in the shape of a long cylinder has a length of $100 \mathrm{m}$, and its mass with occupants is $1000 \mathrm{kg}$. It has strayed too close to a black hole having a mass 100 times that of the Sun (Fig. P11.11). The nose of the spacecraft points toward the black hole, and the distance between the nose and the center of the black hole is $10.0 \mathrm{km} .$ (a) Determine the total force on the spacecraft. (b) What is the difference in the gravitational fields acting on the occupants in the nose of the ship and on those in the rear of the ship, farthest from the black hole? (This difference in accelerations grows rapidly as the ship approaches the black hole. It puts the body of the ship under extreme tension and eventually tears it apart.)

Nathan Nowack

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Problem 12

(a) Compute the vector gravitational field at a point $P$ on the perpendicular bisector of the line joining two objects of equal mass separated by a distance $2 a$ as shown in Figure Pll.12. (b) Explain physically why the field should approach zero as $r \rightarrow 0 .$ (c) Prove mathematically that the answer to part
(a) behaves in this way. (d) Explain physically why the magnitude of the field should approach $2 G M / r^{2}$ as $r \rightarrow \infty$. (e) Prove mathematically that the answer to part (a) behaves correctly in this limit.

Donald Albin

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Problem 13

When a falling meteoroid is at a distance above the Earth's surface of 3.00 times the Earth's radius, what is its acceleration due to the Earth's gravitation?

Mayukh Banik

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Problem 14

Two planets $X$ and $Y$ travel counterclockwise in circular orbits about a star as shown in Figure P11.14. The radii of their orbits are in the ratio $3: 1 .$ At one moment, they are aligned as shown in Figure $\mathrm{P} 11.14 \mathrm{a}$, making a straight line with the star. During the next five years, the angular displacement of planet $\mathrm{X}$ is $90.0^{\circ}$ as shown in Figure $\mathrm{P} 11.14 \mathrm{b}$. What is the angular displacement of planet Y at this moment?

Nathan Nowack

Numerade Educator

Problem 15

A communication satellite in geosynchronous orbit remains above a single point on the Earth's equator as the planet rotates on its axis. (a) Calculate the radius of its orbit. (b) The satellite relays a radio signal from a transmitter near the North Pole to a receiver, also near the North Pole. Traveling at the speed of light, how long is the radio wave in transit?

Nathan Nowack

Numerade Educator

Problem 16

Suppose the Sun's gravity were switched off. The planets would leave their orbits and fly away in straight lines as described by Newton's first law. (a) Would Mercury ever be farther from the Sun than Pluto? (b) If so, find how long it would take Mercury to achieve this passage. If not, give a convincing argument that Pluto is always farther from the Sun than is Mercury.

Nathan Nowack

Numerade Educator

Problem 17

Io, a satellite of Jupiter, has an orbital period of 1.77 days and an orbital radius of $4.22 \times 10^{5} \mathrm{km} .$ From these data, determine the mass of Jupiter.

Donald Albin

Numerade Educator

Problem 18

The Explorer VIII satellite, placed into orbit November 3 $1960,$ to investigate the ionosphere, had the following orbit parameters: perigee, $459 \mathrm{km} ;$ apogee, $2289 \mathrm{km}$ (both distances above the Earth's surface); period, 112.7 min. Find the ratio $v_{p} / v_{a}$ of the speed at perigee to that at apogee.

Nathan Nowack

Numerade Educator

Problem 19

Plaskett's binary system consists of two stars that revolve in a circular orbit about a center of mass midway between them. This statement implies that the masses of the two stars are equal (Fig. P11.19). Assume the orbital speed of each $\operatorname{star}$ is $|\overrightarrow{\mathbf{v}}|=220 \mathrm{km} / \mathrm{s}$ and the orbital period of each is 14.4 days. Find the mass $M$ of each star. (For comparison, the mass of our Sun is $1.99 \times 10^{\$ 0} \mathrm{kg}$.

Nathan Nowack

Numerade Educator

Problem 20

As thermonuclear fusion proceeds in its core, the Sun loses mass at a rate of $3.64 \times 10^{9} \mathrm{kg} / \mathrm{s}$. During the 5000 -yr period of recorded history, by how much has the length of the year changed due to the loss of mass from the Sun? Suggestions: Assume the Earth's orbit is circular. No external torque acts on the Earth-Sun system, so the angular momentum of the Earth is constant.

Donald Albin

Numerade Educator

Problem 21

Comet Halley (Fig. P11.21) approaches the Sun to within $0.570 \mathrm{AU},$ and its orbital period is 75.6 yr. (AU is the symbol for astronomical unit, where $1 \mathrm{AU}=1.50 \times 10^{11} \mathrm{m}$ is the mean Earth-Sun distance.) How far from the Sun will Halley's comet travel before it starts its return journey?

Nathan Nowack

Numerade Educator

Problem 22

How much work is done by the Moon's gravitational field on a 1000 -kg meteor as it comes in from outer space and impacts on the Moon's surface?

Mayukh Banik

Numerade Educator

Problem 23

After the Sun exhausts its nuclear fuel, its ultimate fate will be to collapse to a white dwarf state. In this state, it would have approximately the same mass as it has now, but its radius would be equal to the radius of the Earth. Calculate (a) the average density of the white dwarf, (b) the surface free-fall acceleration, and (c) the gravitational potential energy associated with a $1.00-\mathrm{kg}$ object at the surface of the white dwarf.

Donald Albin

Numerade Educator

Problem 24

A "treetop satellite" moves in a circular orbit just above the surface of a planet, assumed to offer no air resistance. Show that its orbital speed $v$ and the escape speed from the planet are related by the expression $v_{\mathrm{esc}}=\sqrt{2 v}$

Donald Albin

Numerade Educator

Problem 25

An asteroid is on a collision course with Earth. An astronaut lands on the rock to bury explosive charges that will blow the asteroid apart. Most of the small fragments will miss the Earth, and those that fall into the atmosphere will produce only a beautiful meteor shower. The astronaut finds that the density of the spherical asteroid is equal to the average density of the Earth. To ensure its pulverization, she incorporates into the explosives the rocket fuel and oxidizer intended for her return journey. What maximum radius can the asteroid have for her to be able to leave it entirely simply by jumping straight up? On Earth she can jump to a height of $0.500 \mathrm{m}$.

Nathan Nowack

Numerade Educator

Problem 26

A space probe is fired as a projectile from the Earth's surface with an initial speed of $2.00 \times 10^{4} \mathrm{m} / \mathrm{s}$. What will its speed be when it is very far from the Earth? Ignore atmospheric friction and the rotation of the Earth.

Nathan Nowack

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Problem 27

(a) Determine the amount of work that must be done on a 100 -kg payload to elevate it to a height of $1000 \mathrm{km}$ above the Earth's surface. (b) Determine the amount of additional work that is required to put the payload into circular orbit at this elevation.

Nathan Nowack

Numerade Educator

Problem 28

An object is released from rest at an altitude $h$ above the surface of the Earth. (a) Show that its speed at a distance $r$ from the Earth's center, where $R_{E} \leq r \leq R_{E}+h,$ is
$$v=\sqrt{2 G M_{E}\left(\frac{1}{r}-\frac{1}{R_{E}+h}\right)}$$
(b) Assume the release altitude is $500 \mathrm{km} .$ Perform the integral
$$\Delta t=\int_{i}^{f} d t=-\int_{i}^{f} \frac{d r}{v}$$
to find the time of fall as the object moves from the release point to the Earth's surface. The negative sign appears because the object is moving opposite to the radial direction, so its speed is $v=-d r / d t .$ Perform the integral numerically.

Nathan Nowack

Numerade Educator

Problem 29

A satellite of mass $200 \mathrm{kg}$ is placed into Earth orbit at a height of $200 \mathrm{km}$ above the surface. (a) Assuming a circular orbit, how long does the satellite take to complete one orbit? (b) What is the satellite's speed? (c) Starting from the satellite on the Earth's surface, what is the minimum energy input necessary to place this satellite in orbit? Ignore air resistance but include the effect of the planet's daily rotation.

Nathan Nowack

Numerade Educator

Problem 30

A satellite of mass $m$, originally on the surface of the Earth, is placed into Earth orbit at an altitude $h$. (a) Assuming a circular orbit, how long does the satellite take to complete one orbit? (b) What is the satellite's speed? (c) What is the minimum energy input necessary to place this satellite in orbit? Ignore air resistance but include the effect of the planet's daily rotation. Represent the mass and radius of the Earth as $M_{E}$ and $R_{E}$ respectively.

Donald Albin

Numerade Educator

Problem 31

A 500 -kg satellite is in a circular orbit at an altitude of $500 \mathrm{km}$ above the Earth's surface. Because of air friction, the satellite eventually falls to the Earth's surface, where it hits the ground with a speed of $2.00 \mathrm{km} / \mathrm{s}$. How much energy was transformed into internal energy by means of air friction?

Nathan Nowack

Numerade Educator

Problem 32

Derive an expression for the work required to move an Earth satellite of mass $m$ from a circular orbit of radius $2 R_{E}$ to one of radius $3 R_{E}$.

Nathan Nowack

Numerade Educator

Problem 33

A comet of mass $1.20 \times 10^{10} \mathrm{kg}$ moves in an elliptical orbit around the Sun. Its distance from the Sun ranges between $0.500 \mathrm{AU}$ and $50.0 \mathrm{AU}$. (a) What is the eccentricity of its orbit? (b) What is its period? (c) At aphelion, what is the potential energy of the comet-Sun system? Note: $1 \mathrm{AU}=$ one astronomical unit $=$ the average distance from the Sun to the Earth $=1.496 \times 10^{11} \mathrm{m}$.

Donald Albin

Numerade Educator

Problem 34

(a) What is the minimum speed, relative to the Sun, necessary for a spacecraft to escape the solar system if it starts at the Earth's orbit? (b) Voyager 1 achieved a maximum speed of $125000 \mathrm{km} / \mathrm{h}$ on its way to photograph Jupiter. Beyond what distance from the Sun is this speed sufficient to escape the solar system?

Donald Albin

Numerade Educator

Problem 35

(a) A space vehicle is launched vertically upward from the Earth's surface with an initial speed of $8.76 \mathrm{km} / \mathrm{s}$, which is less than the escape speed of $11.2 \mathrm{km} / \mathrm{s}$. What maximum height does it attain? (b) A meteoroid falls toward the Earth. It is essentially at rest with respect to the Earth when it is at a height of $2.51 \times 10^{7} \mathrm{m}$ above the Earth's surface. With what speed does the meteorite (a meteoroid that survives to impact the Earth's surface) strike the Earth?

Nathan Nowack

Numerade Educator

Problem 36

(a) A space vehicle is launched vertically upward from the Earth's surface with an initial speed of $v_{i}$ that is comparable to but less than the escape speed $v_{\mathrm{esc}} .$ What maximum height does it attain? (b) A meteoroid falls toward the Earth. It is essentially at rest with respect to the Earth when it is at a height $h$ above the Earth's surface. With what speed does the meteorite (a meteoroid that survives to impact the Earth's surface) strike the Earth? (c) What If? Assume a baseball is tossed up with an initial speed that is very small compared to the escape speed. Show that the result from part (a) is consistent with Equation 3.15.

Nathan Nowack

Numerade Educator

Problem 37

(a) What value of $n_{i}$ is associated with the 94.96 -nm spectral line in the Lyman series of hydrogen?
(b) What If? Could this wavelength be associated with the Paschen series?
(c) Could this wavelength be associated with the Balmer series?

Zachary Warner

Numerade Educator

Problem 38

A hydrogen atom emits light as it undergoes a transition from the $n=3$ state to the $n=2$ state. Calculate (a) the energy, (b) the wavelength, and (c) the frequency of the radiation.

Nathan Nowack

Numerade Educator

Problem 39

For a hydrogen atom in its ground state, compute (a) the orbital speed of the electron, (b) the kinetic energy of the electron, and (c) the electric potential energy of the atom.

Zachary Warner

Numerade Educator

Problem 40

Show that the speed of the electron in the $n$ th Bohr orbit in hydrogen is given by
$$v_{n}=\frac{k_{e} e^{2}}{n \hbar}$$

Guilherme Barros

Guilherme Barros

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Problem 41

hydrogen atom is in its first excited state $(n=2)$. Calculate
(a) the radius of the orbit,
(b) the linear momentum of the electron,
(c) the angular momentum of the electron,
(d) the kinetic energy of the electron,
(e) the potential energy of the system, and
.(f) the total energy of the system.

Zachary Warner

Numerade Educator

Problem 42

How much energy is required to ionize hydrogen (a) when it is in the ground state and (b) when it is in the state for which $n=3 ?$

Nathan Nowack

Numerade Educator

Problem 43

Two hydrogen atoms collide head-on and end up with zero kinetic energy. Each atom then emits light with a wavelength of $121.6 \mathrm{nm}(n=2 \text { to } n=1 \text { transition }) .$ At what speed were the atoms moving before the collision?

Nathan Nowack

Numerade Educator

Problem 44

A spacecraft of mass $1.00 \times 10^{4} \mathrm{kg}$ is in a circular orbit at an altitude of $500 \mathrm{km}$ above the Earth's surface. Mission Control wants to fire the engines so as to put the spacecraft in an elliptical orbit around the Earth with an apogee of $2.00 \times 10^{4} \mathrm{km} .$ How much energy must be used from the fuel to achieve this orbit? (Assume that all the fuel energy goes into increasing the orbital energy. This model will give a lower limit to the required energy because some of the energy from the fuel will appear as internal energy in the hot exhaust gases and engine parts.)

Nathan Nowack

Numerade Educator

Problem 45

A spacecraft is approaching Mars after a long trip from the Earth. Its velocity is such that it is traveling along a parabolic trajectory under the influence of the gravitational force from Mars. The distance of closest approach will be $300 \mathrm{km}$ above the Martian surface. At this point of closest approach, the engines will be fired to slow down the spacecraft and place it in a circular orbit $300 \mathrm{km}$ above the surface. (a) By what percentage must the speed of the spacecraft be reduced to achieve the desired orbit? (b) How would the answer to part (a) change if the distance of closest approach and the desired circular orbit altitude were $600 \mathrm{km}$ instead of $300 \mathrm{km}$ ? (Note: The energy of the spacecraft-Mars system for a parabolic orbit is $E=0 .)$

Nathan Nowack

Numerade Educator

Problem 46

Astronomers detect a distant meteoroid moving along a straight line that, if extended, would pass at a distance $3 R_{E}$ from the center of the Earth, where $R_{E}$ is the Earth's radius. What minimum speed must the meteoroid have if it is not to collide with the Earth?

Donald Albin

Numerade Educator

Problem 47

Let $\Delta g_{M}$ represent the difference in the gravitational fields produced by the Moon at the points on the Earth's surface nearest to and farthest from the Moon. Find the fraction $\Delta g_{M} / g,$ where $g$ is the Earth's gravitational field. (This difference is responsible for the occurrence of the lunar tides on the Earth.)

Donald Albin

Numerade Educator

Problem 48

Two identical hard spheres, each of mass $m$ and radius $r,$ are released from rest in otherwise empty space with their centers separated by the distance $R$. They are allowed to collide under the influence of their gravitational attraction. (a) Show that the magnitude of the impulse received by each sphere before they make contact is given by $\left[G m^{3}(1 / 2 r-1 / R)\right]^{1 / 2} .$ (b) What If? Find the magnitude of the impulse each receives during their contact if they collide elastically.

Donald Albin

Numerade Educator

Problem 49

Voyager 1 and Voyager 2 surveyed the surface of Jupiter's moon Io and photographed active volcanoes spewing liquid sulfur to heights of $70 \mathrm{km}$ above the surface of this moon. Find the speed with which the liquid sulfur left the volcano. Io's mass is $8.9 \times 10^{22} \mathrm{kg}$, and its radius is $1820 \mathrm{km}$.

Nathan Nowack

Numerade Educator

Problem 50

Two stars of masses $M$ and $m,$ separated by a distance $d,$ revolve in circular orbits about their center of mass (Fig. P11.50). Show that each star has a period given by $$T^{2}=\frac{4 \pi^{2} d^{3}}{G(M+m)}$$
Proceed as follows: Apply Newton's second law to each star. Note that the center-of-mass condition requires that $M r_{2}=m r_{1},$ where $r_{1}+r_{2}=d$.

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 51

A ring of matter is a familiar structure in planetary and stellar astronomy. Examples include Saturn's rings and a ring nebula. Consider a uniform ring of mass $2.36 \times 10^{20} \mathrm{kg}$ and radius $1.00 \times 10^{8} \mathrm{m} .$ An object of mass $1000 \mathrm{kg}$ is placed at a point $A$ on the axis of the ring, $2.00 \times 10^{8}$ $\mathrm{m}$ from the center of the ring (Fig. P11.51). When the object is released, the attraction of the ring makes the object move along the axis toward the center of the ring (point $B$ ).
(a) Calculate the gravitational potential energy of the object-ring system when the object is at $A$
(b) Calculate the gravitational potential energy of the system when the object is at $B .(\mathrm{c})$ Calculate the speed of the object as it passes through $B$.

Nathan Nowack

Numerade Educator

Problem 52

(a) Show that the rate of change of the free-fall acceleration with vertical position near the Earth's surface is
$$\frac{d g}{d r}=-\frac{2 G M_{E}}{R_{E}^{3}}$$
This rate of change with position is called a gradient.
(b) Assuming $h$ is small in comparison to the radius of the Earth, show that the difference in free-fall acceleration between two points separated by vertical distance $h$ is
$$|\Delta g|=\frac{2 G M_{E} h}{R_{E}^{3}}$$
(c) Evaluate this difference for $h=6.00 \mathrm{m},$ a typical height for a two-story building.

Nathan Nowack

Numerade Educator

Problem 53

As an astronaut, you observe a small planet to be spherical. After landing on the planet, you set off, walking always straight ahead, and find yourself returning to your spacecraft from the opposite side after completing a lap of $25.0 \mathrm{km} .$ You hold a hammer and a falcon feather at a height of $1.40 \mathrm{m}$, release them, and observe that they fall together to the surface in 29.2 s. Determine the mass of the planet.

Donald Albin

Numerade Educator

Problem 54

Many people assume air resistance acting on a moving object will always make the object slow down. It can, however, actually be responsible for making the object speed up. Consider a 100 -kg Earth satellite in a circular orbit at an altitude of $200 \mathrm{km} .$ A small force of air resistance makes the satellite drop into a circular orbit with an altitude of $100 \mathrm{km} .$ (a) Calculate the satellite's initial speed. (b) Calculate its final speed in this process. (c) Calculate the initial energy of the satellite-Earth system. (d) Calculate the final energy of the system. (e) Show that the system has lost mechanical energy and find the amount of the loss due to friction. (f) What force makes the satellite's speed increase? Hint: You will find a free-body diagram useful in explaining your answer.

Nathan Nowack

Numerade Educator

Problem 55

The maximum distance from the Earth to the Sun (at aphelion) is $1.521 \times 10^{11} \mathrm{m},$ and the distance of closest approach (at perihelion) is $1.471 \times 10^{11} \mathrm{m}$. The Earth's orbital speed at perihelion is $3.027 \times 10^{4} \mathrm{m} / \mathrm{s}$. Determine (a) the Earth's orbital speed at aphelion and the kinetic and potential energies of the Earth-Sun system (b) at perihelion, and (c) at aphelion. (d) Is the total energy of the system constant? Explain. Ignore the effect of the Moon and other planets.

Donald Albin

Numerade Educator

Problem 56

Two spheres having masses $M$ and $2 M$ and radii $R$ and $3 R,$ respectively, are simultaneously released from rest when the distance between their centers is $12 R$ Assume the two spheres interact only with each other and we wish to find the speeds with which they collide. (a) What $t w o$ isolated system models are appropriate for this system? (b) Write an equation from one of the models and solve it for $\overrightarrow{\mathbf{v}}_{1},$ the velocity of the sphere of mass $M$ at any time after release in terms of $\overrightarrow{\mathbf{v}}_{2},$ the velocity of $2 M .$ (c) Write an equation from the other model and solve it for speed $v_{1}$ in terms of speed $v_{2}$ when the spheres collide. (d) Combine the two equations to find the two speeds $v_{1}$ and $v_{2}$ when the spheres collide.

Dominador Tan

Numerade Educator

Problem 57

Two hypothetical planets of masses $m_{1}$ and $m_{2}$ and radii $r_{1}$ and $r_{2},$ respectively, are nearly at rest when they are an infinite distance apart. Because of their gravitational attraction, they head toward each other on a collision course.
(a) When their center-to-center separation is $d$, find expressions for the speed of each planet and for their relative speed. (b) Find the kinetic energy of each planet just before they collide, taking $m_{1}=2.00 \times 10^{24} \mathrm{kg}, m_{2}=8.00 \times 10^{24} \mathrm{kg}$ $r_{1}=3.00 \times 10^{6} \mathrm{m},$ and $r_{2}=5.00 \times 10^{6} \mathrm{m} .$ Note: Both the energy and momentum of the isolated two-planet system are constant.

Nathan Nowack

Numerade Educator

Problem 58

Why is the following situation impossible? A spacecraft is launched into a circular orbit around the Earth and circles the Earth once an hour.

Donald Albin

Numerade Educator

Problem 59

Assume you are agile enough to run across a horizontal surface at $8.50 \mathrm{m} / \mathrm{s}$, independently of the value of the gravitational field. What would be
(a) the radius and
(b) the mass of an airless spherical asteroid of uniform density $1.10 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$ on which you could launch yourself into orbit by running? (c) What would be your period?
(d) Would your running significantly affect the rotation of the asteroid? Explain.

Donald Albin

Numerade Educator

Problem 60

The oldest artificial satellite still in orbit is Vanguard $I$ launched March $3,1958 .$ Its mass is $1.60 \mathrm{kg} .$ Neglecting atmospheric drag, the satellite would still be in its initial orbit, with a minimum distance from the center of the Earth of $7.02 \mathrm{Mm}$ and a speed at this perigee point of $8.23 \mathrm{km} / \mathrm{s}$ For this orbit, find (a) the total energy of the satellite-Earth system and (b) the magnitude of the angular momentum of the satellite. (c) At apogee, find the satellite's speed and its distance from the center of the Earth. (d) Find the semimajor axis of its orbit. (e) Determine its period.

Nathan Nowack

Numerade Educator

Problem 61

The positron is the antiparticle to the electron. It has the same mass and a positive electric charge of the same magnitude as that of the electron. Positronium is a hydrogenlike atom consisting of a positron and an electron revolving around each other. Using the Bohr model, find (a) the allowed distances between the two particles and (b) the allowed energies of the system.

Nathan Nowack

Numerade Educator

Problem 62

Show that the minimum period for a satellite in orbit around a spherical planet of uniform density $\rho$ is
$$T_{\min }=\sqrt{\frac{3 \pi}{G \rho}}$$
independent of the planet's radius.

Nathan Nowack

Numerade Educator

Problem 63

Studies of the relationship of the Sun to our galaxy-the Milky Way-have revealed that the Sun is located near the outer edge of the galactic disc, about 30000 ly $(1 \mathrm{ly}=9.46 \times$ $10^{15} \mathrm{m}$ ) from the center. The Sun has an orbital speed of approximately $250 \mathrm{km} / \mathrm{s}$ around the galactic center. (a) What is the period of the Sun's galactic motion? (b) What is the order of magnitude of the mass of the Milky Way galaxy?
(c) Suppose the galaxy is made mostly of stars of which the Sun is typical. What is the order of magnitude of the number of stars in the Milky Way?

Nathan Nowack

Numerade Educator

Problem 64

The Solar and Heliospheric Observatory (SOHO) spacecraft has a special orbit, located between the Earth and the Sun along the line joining them, and it is always close enough to the Earth to transmit data easily. Both objects exert gravitational forces on the observatory. It moves around the Sun in a near-circular orbit that is smaller than the Earth's circular orbit. Its period, however, is not less than 1 yr but just equal to 1 yr. Show that its distance from the Earth must be $1.48 \times 10^{9} \mathrm{m} .$ In $1772,$ Joseph Louis Lagrange determined theoretically the special location allowing this orbit. Suggestions: Use data that are precise to four digits. The mass of the Earth is $5.974 \times 10^{24}$ kg. You will not be able to easily solve the equation you generate; instead, use a computer to verify that $1.48 \times 10^{9} \mathrm{m}$ is the correct value.

Nathan Nowack

Numerade Educator

Problem 65

Consider an object of mass $m$, not necessarily small compared with the mass of the Earth, released at a distance of $1.20 \times 10^{7} \mathrm{m}$ from the center of the Earth. Assume the Earth and the object behave as a pair of particles, isolated from the rest of the Universe. (a) Find the magnitude of the acceleration $a_{\mathrm{rel}}$ with which each starts to move relative to the other as a function of $m$. Evaluate the acceleration (b) for $m=5.00 \mathrm{kg},(\mathrm{c})$ for $m=2000 \mathrm{kg},$ and (d) for $m=2.00 \times 10^{24} \mathrm{kg} .(\mathrm{e})$ Describe the pattern of variation of $a_{\text {rel with }} m$.

Donald Albin

Numerade Educator