Fundamentals of Physics, Extended
Fundamentals of Physics, Extended
12th Edition
ISBN: 9781119773474
Author: David Halliday; Robert Resnick; Jearl Walker
Publisher: Wiley Global Education US
Question
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Chapter 13, Problem 96P

(a)

To determine

To calculate: Compactness of Earth.

Compactness of Earth is 1.383×109 .

Given:

Mass of Earth: 5.97×1024 kg .

Radius of Earth: 6.4×106 m .

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius Rs=2GMc2 where M is the mass of body.

Calculation:

Let compactness be denoted by Z and for earth ZEarth=RsREarth

Substituting the given values we get-

  ZEarth=2GMearthc2RearthZEarth=2×6.67×1011m3kg1s2×5.97×1024kg3×108ms12×6.4×106mZEarth=1.383×109

Conclusion:

Compactness of Planet earth is 1.383×109 .

(b)

To calculate: Compactness of Sun.

Compactness of Sun is 4.238×106 .

Given:

Mass of Sun: 1.99×1030 kg .

Radius of Sun: 6.96×108 m .

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius Rs=2GMc2 where M is the mass of body.

Calculation:

Let compactness be denoted by Z and for earth ZSun=RsRSun

Substituting the given values we get-

  ZSun=2GMSunc2RSunZSun=2×6.67×1011m3kg1s2×1.99×1030kg3×108ms12×6.96×108mZSun=4.238×106

Conclusion:

Compactness of Sun is 4.238×106 .

(c)

To calculate: Compactness of Neutron Star.

Compactness of Neutron Star is 0.9934 .

Given:

Density of Star: 4×1017 kg/m3

Radius of Star: 2×104 m

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius Rs=2GMc2 where M is the mass of body.

Calculation:

Let compactness be denoted by Z and for earth ZStar=RsRStar

Substituting the given values we get-

  ZStar=2GMStarc2RStar

Since ρ=MV then MStar=43πρRStar3

  ZStar=8G3c2πρRStar2ZStar=8×6.67×1011m3kg1s2×π×4×1017kg m3×2×104m23×3×108ms12ZStar=0.9934

Conclusion:

Compactness of Star is 0.9934 .

(d)

To calculate: Compactness of Black hole.

Compactness of Black hole is1.

Given:

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius Rs=2GMc2 where M is the mass of body.

Calculation:

Let compactness be denoted by Z and for earth ZBlackhole=RsRBlackhole

According to question the Schwarzschild radius of black hole is comparable to its actual radius therefore RsR

  ZBlackhole=RRZBlackhole=1

Conclusion:

Compactness of Black hole is 1 .

(b)

To determine

To calculate: Compactness of Sun.

Compactness of Sun is 4.238×106 .

Given:

Mass of Sun: 1.99×1030 kg .

Radius of Sun: 6.96×108 m .

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius Rs=2GMc2 where M is the mass of body.

Calculation:

Let compactness be denoted by Z and for earth ZSun=RsRSun

Substituting the given values we get-

  ZSun=2GMSunc2RSunZSun=2×6.67×1011m3kg1s2×1.99×1030kg3×108ms12×6.96×108mZSun=4.238×106

Conclusion:

Compactness of Sun is 4.238×106 .

(c)

To calculate: Compactness of Neutron Star.

Compactness of Neutron Star is 0.9934 .

Given:

Density of Star: 4×1017 kg/m3

Radius of Star: 2×104 m

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius Rs=2GMc2 where M is the mass of body.

Calculation:

Let compactness be denoted by Z and for earth ZStar=RsRStar

Substituting the given values we get-

  ZStar=2GMStarc2RStar

Since ρ=MV then MStar=43πρRStar3

  ZStar=8G3c2πρRStar2ZStar=8×6.67×1011m3kg1s2×π×4×1017kg m3×2×104m23×3×108ms12ZStar=0.9934

Conclusion:

Compactness of Star is 0.9934 .

(d)

To calculate: Compactness of Black hole.

Compactness of Black hole is1.

Given:

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius Rs=2GMc2 where M is the mass of body.

Calculation:

Let compactness be denoted by Z and for earth ZBlackhole=RsRBlackhole

According to question the Schwarzschild radius of black hole is comparable to its actual radius therefore RsR

  ZBlackhole=RRZBlackhole=1

Conclusion:

Compactness of Black hole is 1 .

(c)

To determine

To calculate: Compactness of Neutron Star.

Compactness of Neutron Star is 0.9934 .

Given:

Density of Star: 4×1017 kg/m3

Radius of Star: 2×104 m

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius Rs=2GMc2 where M is the mass of body.

Calculation:

Let compactness be denoted by Z and for earth ZStar=RsRStar

Substituting the given values we get-

  ZStar=2GMStarc2RStar

Since ρ=MV then MStar=43πρRStar3

  ZStar=8G3c2πρRStar2ZStar=8×6.67×1011m3kg1s2×π×4×1017kg m3×2×104m23×3×108ms12ZStar=0.9934

Conclusion:

Compactness of Star is 0.9934 .

(d)

To calculate: Compactness of Black hole.

Compactness of Black hole is1.

Given:

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius Rs=2GMc2 where M is the mass of body.

Calculation:

Let compactness be denoted by Z and for earth ZBlackhole=RsRBlackhole

According to question the Schwarzschild radius of black hole is comparable to its actual radius therefore RsR

  ZBlackhole=RRZBlackhole=1

Conclusion:

Compactness of Black hole is 1 .

(d)

To determine

To calculate: Compactness of Black hole.

Compactness of Black hole is1.

Given:

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius Rs=2GMc2 where M is the mass of body.

Calculation:

Let compactness be denoted by Z and for earth ZBlackhole=RsRBlackhole

According to question the Schwarzschild radius of black hole is comparable to its actual radius therefore RsR

  ZBlackhole=RRZBlackhole=1

Conclusion:

Compactness of Black hole is 1 .

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Chapter 13 Solutions

Fundamentals of Physics, Extended

Ch. 13 - Mountain pull. A large mountain can slightly...Ch. 13 - SSM At what altitude above Earths surface would...Ch. 13 - Mile-high building. In 1956, Frank Lloyd Wright...Ch. 13 - ILW Certain neutron stars extremely dense stars...Ch. 13 - Two concentric spherical shells with uniformly...Ch. 13 - A solid sphere has a uniformly distributed mass of...Ch. 13 - Prob. 26PCh. 13 - In Problem 1, what ratio m/M gives the least...Ch. 13 - SSM The mean diameters of Mars and Earth are 6.9 ...Ch. 13 - a What is the gravitational potential energy of...Ch. 13 - Prob. 33PCh. 13 - Zero, a hypothetical planet, has a mass of 5.0 ...Ch. 13 - In deep space, sphere A of mass 20 kg is located...Ch. 13 - Prob. 39PCh. 13 - A projectile is shot directly away from Earths...Ch. 13 - SSM Two neutron stars arc separated by a distance...Ch. 13 - a What linear speed must an Earth satellite have...Ch. 13 - Prob. 44PCh. 13 - The Martian satellite Photos travels in an...Ch. 13 - The first known collision between space debris and...Ch. 13 - Prob. 47PCh. 13 - The mean distance of Mars from the Sun is 1.52...Ch. 13 - Prob. 49PCh. 13 - Prob. 50PCh. 13 - Prob. 51PCh. 13 - The Suns center is at one focus of Earths orbit....Ch. 13 - A 20 kg satellite has a circular orbit with a...Ch. 13 - In 1610, Galileo used his telescope to discover...Ch. 13 - In 1993 the spacecraft Galileo sent an image Fig....Ch. 13 - Prob. 57PCh. 13 - Three identical stars of mass M form an...Ch. 13 - Prob. 61PCh. 13 - Prob. 62PCh. 13 - SSM WWW An asteroid, whose mass is 2.0 10-4 times...Ch. 13 - A satellite orbits a planet of unknown mass in a...Ch. 13 - A Satellite is in a circular Earth orbit of radius...Ch. 13 - One way to attack a satellite in Earth orbit is to...Ch. 13 - Prob. 67PCh. 13 - Prob. 70PCh. 13 - Prob. 72PCh. 13 - The mysterious visitor that appears in the...Ch. 13 - ILW The masses and coordinates of three spheres...Ch. 13 - SSM A very early, simple satellite consisted of an...Ch. 13 - GO Four uniform spheres, with masses mA = 40 kg,...Ch. 13 - a In Problem 77, remove sphere A and calculate the...Ch. 13 - Prob. 80PCh. 13 - Prob. 81PCh. 13 - Prob. 82PCh. 13 - Prob. 83PCh. 13 - Prob. 84PCh. 13 - Prob. 85PCh. 13 - Prob. 86PCh. 13 - Prob. 87PCh. 13 - Prob. 88PCh. 13 - Prob. 89PCh. 13 - Prob. 90PCh. 13 - Prob. 91PCh. 13 - Prob. 92PCh. 13 - Prob. 93PCh. 13 - Prob. 94PCh. 13 - Prob. 95PCh. 13 - Prob. 96P
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