Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th Edition)

Published by Pearson
ISBN 10: 0133942651
ISBN 13: 978-0-13394-265-1

Chapter 13 - Newton's Theory of Gravity - Exercises and Problems: 41

Answer

The distance from the black hole to the center of the earth is $1999~R_e$ (where $R_e$ is the earth's radius).

Work Step by Step

We can write an expression force the force of gravity $F_e$ exerted by the earth on an object of mass $M$ at the earth's surface. Let $M_e$ be the earth's mass and let $R_e$ be the earth's radius. $F_e = \frac{G~M_e~M}{R_e^2}$ We can write an expression force the force of gravity $F_b$ exerted by the black hole on an object of mass $M$ at the earth's surface. Let $M_b$ be the black hole's mass and let $R_b$ be the distance from the black hole. $F_b = \frac{G~M_b~M}{R_b^2}$ Objects on the earth's surface will begin to lift off the surface when $F_b$ is greater than $F_e$. We can equate $F_b$ and $F_e$ to find the distance $R_b$ from the earth's surface when this would begin to happen. $F_e = F_b$ $\frac{G~M_e~M}{R_e^2} = \frac{G~M_b~M}{R_b^2}$ $R_b^2 = \frac{M_b~R_e^2}{M_e}$ $R_b = \sqrt{\frac{M_b}{M_e}}~R_e$ $R_b = \sqrt{\frac{(12)(1.99\times 10^{30}~kg)}{5.98\times 10^{24}~kg}}~R_e$ $R_b = 1998~R_e$ The distance from the black hole to the earth's surface is $1998~R_e$. Therefore, the distance from the black hole to the center of the earth is $1999~R_e$.
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