Explanation 1) Concept: We can equate the gravitational force between the planet and the star to the centripetal force . Put the distance of the star (r 1 ) from C.O.M in this equation. Using the period of the star, we can find the mass m 2 of the dark star. Comparing it with the mass of the Sun we can write it in terms of M s. 2) Given: i) The velocity of the visible star is v = 270 k m s = 2.7 × 10 5 m s . ii) The orbital period of the visible star is, T = 1.70 d a y s = 1.70 86400 = 146880 s . iii) The approximate mass of the visible star is, m 1 = 6 M s . iv) The mass of the sun is, M s = 1.99 × 10 30 k g . 3) Formulae: i) The Centripetal force F = M v 2 r ii) The Gravitational force of attraction between two bodies of masses M and m separated by distance d is, F = G M m r 2 4) Calculations: The binary star system is orbiting about their center of mass. The gravitational force of attraction between them provides the centripetal force to keep their motion in the circular orbit. So we have, F = G m 1 m 2 r 2 = m 1 v 2 r 1 As both stars revolves about the center of mass, r 1 = ( m 1 0 + m 2 r ) m 1 + m 2 r 1 = m 2 r m 1 + m 2 r = m 1 + m 2 m 2 r 1 The orbital speed of m 1 in terms of T is written as, v = 2 π r 1 T So, r 1 = v T 2 π Substituting r 1 in r, r = m 1 + m 2 m 2 v T 2 π Putting r and r 1 in F 1 , F = G m 1 m 2 ( m 1 + m 2 ) m 2 v T 2 π 2 = m 1 v 2 v T 2 π F =

10th Edition

ISBN: 9781118230725

Author: David Halliday, Robert Resnick, Jearl Walker

Publisher: Wiley, John & Sons, Incorporated

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Chapter 13, Problem 54P

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The approximate mass m₂ of the dark star as integral multiple of M_s

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Chapter 13, Problem 53P

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Chapter 13 Solutions

Fundamentals of Physics Extended

Ch. 13 - In Fig. 13-21, a central particle of mass M is...Ch. 13 - Prob. 2Q Ch. 13 - In Fig. 13-23, a central particle is surrounded by...Ch. 13 - In Fig. 13-24, two particles, of masses m and 2m,...Ch. 13 - Prob. 5Q Ch. 13 - In Fig. 13-26, three particles are fixed in place....Ch. 13 - Rank the four systems of equal- mass particles...Ch. 13 - Figure 13-27 gives the gravitational acceleration...Ch. 13 - Figure 13-28 shows three particles initially fixed...Ch. 13 - Figure 13-29 shows six paths by which a rocket...

Ch. 13 - Figure 13-30 shows three uniform spherical planets...Ch. 13 - In Fig. 13-31, a particle of mass m which is not...Ch. 13 - ILW A mass M is split into two parts, m and M m,...Ch. 13 - Moon effect. Some people believe that the Moon...Ch. 13 - Prob. 3P Ch. 13 - The Sun and Earth each exert a gravitational force...Ch. 13 - Miniature black holes. Left over from the big-bang...Ch. 13 - GO In Fig. 13-32, a square of edge length 20.0 cm...Ch. 13 - One dimension. In Fig. 13-33, two point particles...Ch. 13 - In Fig. 13-34, three 5.00 kg spheres are located...Ch. 13 - SSM WWW We want to position a space probe along a...Ch. 13 - Prob. 10P Ch. 13 - As seen in Fig. 13-36, two spheres of mass m and a...Ch. 13 - GO In Fig. 13-37a, particle A is fixed in place at...Ch. 13 - Figure 13-38 shows a spherical hollow inside a...Ch. 13 - Prob. 14P Ch. 13 - GO Three dimensions. Three point particles are...Ch. 13 - GO In Fig. 13-40, a particle of mass m1 = 0.67 kg...Ch. 13 - a What will an object weigh on the Moons surface...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 - Prob. 22P Ch. 13 - Prob. 23P Ch. 13 - Two concentric spherical shells with uniformly...Ch. 13 - A solid sphere has a uniformly distributed mass of...Ch. 13 - Prob. 26P Ch. 13 - Figure 13-42 shows, not to scale, a cross section...Ch. 13 - Prob. 28P Ch. 13 - Prob. 29P Ch. 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. 33P Ch. 13 - Prob. 34P Ch. 13 - GO Figure 13-44 shows four particles, each of mass...Ch. 13 - Zero, a hypothetical planet, has a mass of 5.0 ...Ch. 13 - GO The three spheres in Fig, 13-45, with masses mA...Ch. 13 - In deep space, sphere A of mass 20 kg is located...Ch. 13 - Prob. 39P Ch. 13 - A projectile is shot directly away from Earths...Ch. 13 - SSM Two neutron stars arc separated by a distance...Ch. 13 - GO Figure 13-46a shows a particle A that can he...Ch. 13 - a What linear speed must an Earth satellite have...Ch. 13 - Prob. 44P Ch. 13 - The Martian satellite Photos travels in an...Ch. 13 - The first known collision between space debris and...Ch. 13 - Prob. 47P Ch. 13 - The mean distance of Mars from the Sun is 1.52...Ch. 13 - Prob. 49P Ch. 13 - Prob. 50P Ch. 13 - Prob. 51P Ch. 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 - Prob. 54P Ch. 13 - In 1610, Galileo used his telescope to discover...Ch. 13 - In 1993 the spacecraft Galileo sent an image Fig....Ch. 13 - Prob. 57P Ch. 13 - Prob. 58P Ch. 13 - Three identical stars of mass M form an...Ch. 13 - In Fig. 13-50, two satellites, A and B, both of...Ch. 13 - Prob. 61P Ch. 13 - Prob. 62P Ch. 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. 67P Ch. 13 - GO Two small spaceships, each with mass m = 2000...Ch. 13 - Prob. 69P Ch. 13 - Prob. 70P Ch. 13 - Several planets Jupiter. Saturn, Uranus are...Ch. 13 - Prob. 72P Ch. 13 - Figure 13-53 is a graph of the kinetic energy K of...Ch. 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. 79P Ch. 13 - Prob. 80P Ch. 13 - Prob. 81P Ch. 13 - Prob. 82P Ch. 13 - Prob. 83P Ch. 13 - Prob. 84P Ch. 13 - Prob. 85P Ch. 13 - Prob. 86P Ch. 13 - Prob. 87P Ch. 13 - Prob. 88P Ch. 13 - Prob. 89P Ch. 13 - A 50 kg satellite circles planet Cruton every 6.0...Ch. 13 - Prob. 91P Ch. 13 - A 150.0 kg rocket moving radially outward from...Ch. 13 - Prob. 93P Ch. 13 - Two 20 kg spheres are fixed in place on a y axis,...Ch. 13 - Sphere A with mass 80 kg is located at the origin...Ch. 13 - In his 1865 science fiction novel From the Earth...Ch. 13 - Prob. 97P Ch. 13 - Prob. 98P Ch. 13 - A thin rod with mass M = 5.00 kg is bent in a...Ch. 13 - In Fig. 13-57, identical blocks with identical...Ch. 13 - A spaceship is on a straight-line path between...

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