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MasteringPhysics: Assignment Print View Test Your Understanding 12.2: Surface Gravity The earth has mass and radius . The acceleration due to gravity at the earth's surface, also called the earth's surface gravity, is . Part A Rank the following planets in order of their mass, from largest value to smallest value. ANSWER: View All attempts used; correct answer displayed If a planet has mass and radius , the planet's surface gravity is We can rewrite this as an expression for the planet's mass: For the earth, this says Use this to calculate the mass for each of the five planets: : = : http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=1484636 (1 of 23) [12/13/2010 7:18:13 PM]

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Test Your Understanding 12.2: Surface Gravity

The earth has mass and radius . The acceleration due to gravity at the earth's surface, also called the earth's surface gravity, is .

Part A

Rank the following planets in order of their mass, from largest value to smallest value.

ANSWER:

View All attempts used; correct answer displayed

If a planet has mass and radius , the planet's surface gravity is

We can rewrite this as an expression for the planet's mass:

For the earth, this says

Use this to calculate the mass for each of the five planets: :

=

:

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=

:

=

:

=

:

=

Problem 12.88: Tidal Forces near a Black Hole

An astronaut inside a spacecraft, which protects her from harmful radiation, is orbiting a black hole at a distance of 120 from its center. The black hole is 5.00 times the mass of the sun and has a Schwarzschild radius of 15.0 . The astronaut is positioned inside the spaceship such that one of her 0.030 ears is 6.0 farther from the black hole than the center of mass of the spacecraft and the other ear is 6.0 closer.

Part AWhat is the tension between her ears?

Express your answer using two significant figures.

ANSWER: = 2100

Correct

Part B

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Would the astronaut find it difficult to keep from being torn apart by the gravitational forces? (Since her whole body orbits with the same angular velocity, one ear is moving too slowly for the radius of its orbit and the other is moving too fast. Hence her head must exert forces on her ears to keep them in their orbits.)

ANSWER: My Answer:

Part CIs the center of gravity of her head at the same point as the center of mass?

ANSWER: My Answer:

Exercise 12.20

Neutron stars, such as the one at the center of the Crab Nebula, have about the same mass as our sun but a much smaller diameter.

Part A

If you weigh 675 on the earth, what would you weigh when at the surface of a neutron star that has the same mass as our sun and a diameter of 17 ?

Express your answer using two significant figures.

ANSWER: = 1.3×1014

Correct

Problem 12.75

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Part AComets travel around the sun in elliptical orbits with large eccentricities. If a comet has speed 2.4×104 when at a distance of 2.7×1011 from the center of the sun, what is its speed when at a distance of 6.0×1010 .

Express your answer using two significant figures.

ANSWER: = 6.3×104

Correct

Problem 12.79

A 3250-kg spacecraft is in a circular orbit a distance 1620 above the surface of Mars.

Part AHow much work must the spacecraft engines perform to move the spacecraft to a circular orbit that is 3720 above the surface?

ANSWER: = 4.09×109

Correct

Exercise 12.2: Cavendish Experiment

In the Cavendish balance apparatus shown in the figure12.4 in the textbook, suppose that = 1.50 , = 23.0 , and the rod connecting the pairs is 29.0 long. In each

pair, and are 13.0 apart center-to-center.

Part AFind the net force.

ANSWER: = 0×100

Correct

Part B

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Find the net the net torque (about the rotation axis) on the rotating part of the apparatus.

ANSWER: =

3.08×10−8 All attempts used; correct answer displayed

Part CSuggest some ways to improve the sensitivity of this experiment.

ANSWER: My Answer:

Exercise 13.40

When displaced from equilibrium, the two hydrogen atoms in an molecule are acted on

by a restoring force with .

Part A

Calculate the oscillation frequency of the molecule. (Hint: The mass of a hydrogen

atom is 1.008 atomic mass units, or 1 u; .)

ANSWER: = 1.33×1014

Correct

Problem 13.93

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Two identical, thin rods, each with mass and length , are joined at right angles to form an L-shaped object. This object is balanced on top of a sharp edge . If the L-shaped object is deflected slightly, it oscillates.

Part AFind the frequency of oscillation.

Express your answer in terms of the variables , and .

ANSWER:

=

Correct

Test Your Understanding 13.6: The Physical Pendulum

A physical pendulum is undergoing angular oscillations around its equilibrium position, swinging from left to right and from right to left.

Part A

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At the instant that the center of gravity of the pendulum is at the equilibrium position and moving from left to right, the acceleration of the center of gravity is

ANSWER: to the left downwarddownward and to the left upwardupward and to the left upward and to the rightzero to the rightdownward and to the right

Correct

At the instant in question, the center of gravity is neither speeding up nor slowing down. Hence the acceleration has no component along the direction of motion (to the left or right). The center of gravity is following a circular path centered on the pivot, which is above the center of gravity. Hence the acceleration of the center of gravity is upward, toward the pivot.

Exercise 13.20

An object is undergoing SHM with period 0.315 and amplitude 6.10 . At the object is instantaneously at rest at 6.10 .

Part ACalculate the time it takes the object to go from 6.10 to -1.60 .

ANSWER: = 9.21×10−2

Correct

Problem 13.67

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A 1.50 , horizontal, uniform tray is attached to a vertical ideal spring of force constant 200

and a 255 metal ball is in the tray. The spring is below the tray, so it can oscillate up-and-down. The tray is then pushed down 14.2 below its equilibrium point (call this point ) and released from rest.

Part A

How high above point will the tray be when the metal ball leaves the tray? (Hint: This does not occur when the ball and tray reach their maximum speeds.)

ANSWER: = 22.8

Correct

Part B

How much time elapses between releasing the system at point and the ball leaving the tray?

ANSWER: = 0.208

Correct

Part CHow fast is the ball moving just as it leaves the tray?

ANSWER: = 1.21

Correct

Problem 13.72

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An object with mass 0.210 is acted on by an elastic restoring force with force

constant 10.7 . The object is set into oscillation with an initial potential energy of 0.140

and an initial kinetic energy of 6.40×10−2 .

Part AWhat is the amplitude of oscillation?

ANSWER: = 0.195 Correct

Part BWhat is the potential energy when the displacement is one-half the amplitude?

ANSWER: = 5.10×10−2 Correct

Part CAt what displacement are the kinetic and potential energies equal?

ANSWER: = 0.138 Correct

Part D

What is the value of the phase angle if the initial velocity is positive and the initial displacement is negative?

ANSWER: =

3.74 All attempts used; correct answer displayed

Problem 15.84: Instantaneous Power in a Standing Wave

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The instantaneous rate at which a wave transmits energy along a string (instantaneous power) is

where is the tension.

Part A

Evaluate for a standing wave of the form given by equation

.

Express your answer in terms of the variables , , , , , and .

ANSWER: =

Correct

Part B

Show that for all values of , the average power carried by the standing wave is zero.

(Equation does not apply here. Can you see why?)

ANSWER: My Answer:

Part C

For a standing wave given by equation , graph as

functions of for . (Positive means energy is flowing in the -

direction negative means the flow is in the -direction.)

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ANSWER:

View All attempts used; correct answer displayed

Part D

For a standing wave given by equation , graph the

displacement as functions of for .

ANSWER:

View All attempts used; correct answer displayed

Part E

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For a standing wave given by equation , graph as

functions of for . (Positive means energy is flowing in the -

direction negative means the flow is in the -direction.)

ANSWER:

View Correct

Part F

For a standing wave given by equation , graph the

displacement as functions of for .

ANSWER:

View All attempts used; correct answer displayed

Part G

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For a standing wave given by equation , graph as

functions of for . (Positive means energy is flowing in the -

direction negative means the flow is in the -direction.)

ANSWER:

View Answer Requested

Part H

For a standing wave given by equation , graph the

displacement as functions of for .

ANSWER:

View Answer Requested

Part I

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The kinetic energy per unit length of string is greatest where the string has the greatest transverse speed, and the potential energy per unit length of string is greatest where the string has the steepest slope (because there the string is stretched the most). Using these ideas, discuss the flow of energy along the string.

ANSWER: My Answer:

Exercise 15.44

One string of a certain musical instrument is 72.0 long and has a mass of 8.76 . It is being played in a room where the speed of sound is 344 .

Part ATo what tension must you adjust the string so that, when vibrating in its second overtone, it produces sound of wavelength 3.40 ?

ANSWER: = 2.87×105 Correct

Part BWhat frequency sound does this string produce in its fundamental mode of vibration?

ANSWER: = 3370 Correct

Problem 15.60

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A vertical, 1.26- length of 18-gauge (diameter of 1.024 ) copper wire has a 120.0- ball hanging from it.

Part AWhat is the wavelength of the third harmonic for this wire?

ANSWER: = 0.840

Correct

Part B

A 600.0- ball now replaces the original ball. What is the change in the wavelength of the third harmonic caused by replacing the light ball with the heavy one? (Hint: See Table 11.1 in the textbook for Young's modulus.)

Express your answer using two significant figures.

ANSWER: = 4.4

Correct

Problem 15.70

A guitar string is vibrating in its fundamental mode, with nodes at each end. The length of the segment of the string that is free to vibrate is 0.389 . The maximum transverse acceleration of a point at the middle of the segment is 8000 and the maximum

transverse velocity is 3.10 .

Part AWhat is the amplitude of this standing wave?

ANSWER: = 1.20×10−3 Correct

Part B

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What is the wave speed for the transverse traveling waves on this string?

ANSWER: = 320 Correct

Problem 15.75

A uniform cylindrical steel wire, 54.0 long and 1.12 in diameter, is fixed at both ends.

Part ATo what tension must it be adjusted so that, when vibrating in its first overtone, it produces the note of frequency 311 ? Assume that it stretches an insignificant amount. (Hint:

The density of the steel is 7800 .)

Express your answer using two significant figures.

ANSWER: = 220

Correct

Problem 15.82

A deep-sea diver is suspended beneath the surface of Loch Ness by a cable of length = 100 that is attached to a boat on the surface . The diver and his suit have a total

mass of = 120 and a volume

of = 7.80×10−2 . The cable has a diameter of 2.35 and a linear mass density of = 1.01 . The diver thinks he sees something moving in the murky depths and jerks the end of the cable back and forth to send transverse waves up the cable as a signal to his companions in the boat.

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Part AWhat is the tension in the cable at its lower end, where it is attached to the diver? Do not forget to include the buoyant force that the water (density = 1000 ) exerts on him.

ANSWER: = 412

Correct

Part BCalculate the tension in the cable a distance above the diver. The buoyant force on the cable must be included in your calculation.

Express your answer in terms of the variables , , , , , , and appropriate constants.

ANSWER:

=

Answer Requested

Part C

The speed of transverse waves on the cable is given by . The speed therefore

varies along the cable, since the tension is not constant. (This expression neglects the damping force that the water exerts on the moving cable.) Integrate to find the time required for the first signal to reach the surface.

ANSWER: = 3.90

Correct

Test Your Understanding 16.4: Standing Sound Waves and

Normal Modes

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The first four normal-mode frequencies of a pipe open at both ends are 100 , 200 , 300 , and 400 .

Part AIf you now close one end of this pipe, how many of these four frequencies are still normal-mode frequencies of the pipe?

ANSWER: only one of these is still a normal-mode frequencynone are still normal-mode frequenciesall four are still normal-mode frequenciesjust two of these are still normal-mode frequenciesjust three of these are still normal-mode frequencies

Correct

When the pipe is open at both ends, its normal-mode frequencies are

so . When one end is closed, the lowest (fundamental) normal-mode frequency becomes

and the normal-mode frequencies are

Hence after one end of the pipe is closed, none of the frequencies 100 , 200 , 300 , and 400 are still normal-mode frequencies.

Test Your Understanding 16.5: Resonance and Sound

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An open pipe has length 60.0 . You wish to place a second, stopped pipe next to the open pipe. You want to choose the stopped pipe so that when the air in the stopped pipe oscillates at its fundamental frequency, it causes the air in the open pipe to vibrate at its third-harmonic frequency.

Part AWhat must be the length of the stopped pipe?

ANSWER: 80.0

360 10.0

30.0

40.0

20.0

120 180 90.0

Correct

Let be the length of the stopped pipe (which is our target variable). The open pipe

has length . The fundamental frequency of the stopped pipe is

and the third-harmonic frequency of the open pipe is

In both of these expressions is the speed of sound in air, which is the same in both pipes. To achieve a resonance between these two modes of vibration, the frequencies must be the same:

If we solve this for , we get

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Exercise 16.34

Two loudspeakers, A and B (the figure ), are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 2.00 to the right of speaker A. The frequency of the sound waves produced by the loudspeakers is 206 . Consider point P between the speakers and along the line connecting them, a distance x to the right of speaker A. Both speakers emit sound waves that travel directly from the speaker to point P.

Part AFor what values of x will destructive interference occur at point P?

Enter your answers numerically separated by a comma. Express your answers using two significant figures.

ANSWER: , = 0.58,1.4

Correct

Part BFor what values of x will constructive interference occur at point P?

Enter your answers numerically separated by commas. Express your answers using two significant figures.

ANSWER: , , = 1.0,0.17,1.8

Correct

Part C

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Interference effects like those in parts A and B are almost never a factor in listening to home stereo equipment. Why not?

ANSWER: My Answer:

Problem 16.60: A New Musical Instrument

You have designed a new musical instrument of very simple construction. Your design consists of a metal tube with length and diameter . You have stretched a string of mass per unit length across the open end of the tube. The other end of the tube is closed. To produce the musical effect you're looking for, you want the frequency of the third-harmonic standing wave on the string to be the same as the fundamental frequency for sound waves in the air column in the tube. The speed of sound waves in this air column is

.

Part AWhat must be the tension of the string to produce the desired effect?

Express your answer in terms of the given quantities.

ANSWER: =

Correct

Part BWhat happens to the sound produced by the instrument if the tension is changed to twice the value calculated in part (a)?

ANSWER: My Answer:

Part CFor the tension calculated in part (a), what other harmonics of the string, if any, are in resonance with standing waves in the air column?

ANSWER: My Answer:

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Exercise 16.49: Doppler Radar

A giant thunderstorm is moving toward a weather station at 47.0 .

Part A

If the station sends a radar beam of frequency 215.0 toward the storm, what is the difference in frequency between the emitted beam and the beam reflected back from the storm? Be careful to carry plenty of significant figures! (Hint: The storm reflects the same frequency that it receives.)

ANSWER: = 30.1

Correct

Exercise 16.51

Part AHow fast (as a percentage of light speed) would a star have to be moving so that the frequency of the light we receive from it is 15.0 higher than the frequency of the light it is emitting?

ANSWER: =

0.139 All attempts used; correct answer displayed

Part BWould it be moving away from us or toward us? (Assume it is moving either directly away from us or directly toward us.)

ANSWER: away from ustoward us

Correct

Problem 16.74

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A 2.45- sound wave travels through a pregnant woman’s abdomen and is reflected from the fetal heart wall of her unborn baby. The heart wall is moving toward the sound receiver as the heart beats. The reflected sound is then mixed with the transmitted sound, and 85 beats per second are detected. The speed of sound in body tissue is 1490 .

Part ACalculate the speed of the fetal heart wall at the instant this measurement is made.

ANSWER: = 2.58×10−2

Correct

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