Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

A satellite orbits a planet of unknown mass in a circle of radius $2.0 \times 10^{7} m$ . The magnitude of the gravitational force on the satellite from the planet is $F = 80 N$ .
(a) What is the kinetic energy of the satellite in this orbit?
(b) What would F be if the orbit radius were increased to $3.0 \times 10^{7} m$ ?

Kinetic energy of the satellite $K = 8.0 \times 10^{8} J$
Gravitational force when the orbit radius is increased to $3 \times 10^{7} m is (F_{2}) = 35.5 N$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Listing the given quantities

$r_{1} = 2 \times 10^{7} m$

$F_{1} = 80 N$

$r_{2} = 3 \times 10^{7} m$

Step 2: Understanding the concept of gravitational force and kinetic energy

Using the given gravitational force and kinetic energy formula, we can find the kinetic energy of the satellite ( $K_{s}$ ). Using $F \propto \frac{1}{r^{2}},$ we can find gravitational force $F_{2}$ when the radius is increased to ${3×10}^{7} m$

Formula:

$K= \frac{GMm}{2 (r)}$

$F= \frac{GMm}{r^{2}}$

Step 3: (a) Calculation of kinetic energy of satellite

$K= \frac{GMm}{2 (r_{1})}$

$F_{1} = \frac{GMm}{r_{1}^{2}}$

$F_{1} {×r}_{1} = \frac{GMm}{r_{1}}$

Using this in kinetic energy equation (1),

$\begin{matrix} K = \frac{1}{2} F_{1} \times r_{1} \\ = \frac{1}{2} 80 N \times 2 \times 10^{7} m \\ = 8.0 \times 10^{8} J \end{matrix}$

Step 4: (b) Calculation of Gravitational force when the orbit radius is increased

$\begin{matrix} F_{1} \propto \frac{1}{r_{1}^{2}} \\ F_{2} \propto \frac{1}{r_{2}^{2}} \\ \frac{F_{1}}{F_{2}} = \frac{r_{2}^{2}}{r_{1}^{2}} \\ \frac{F_{1}}{F_{2}} = \frac{{(2 \times 10^{7} m)}^{2}}{{(3 \times 10^{7} m)}^{2}} \\ \frac{F_{1}}{F_{2}} = \frac{4}{9} \end{matrix}$

Using the given value of $F_{1}$

$\begin{matrix} F_{2} = \frac{4 \times 80 N}{9} \\ = 35.5 N \end{matrix}$

Gravitational force when the orbit radius is increased to $3 \times 10^{7} m is F_{2} = 35.5 N$

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: Listing the given quantities

Step 2: Understanding the concept of gravitational force and kinetic energy

Step 3: (a) Calculation of kinetic energy of satellite

Step 4: (b) Calculation of Gravitational force when the orbit radius is increased

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Fundamentals Of Physics

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Short Answer

A satellite orbits a planet of unknown mass in a circle of radius 2.0×107 m. The magnitude of the gravitational force on the satellite from the planet is F=80 N. (a) What is the kinetic energy of the satellite in this orbit? (b) What would F be if the orbit radius were increased to 3.0×107 m ?

Step by Step Solution

Step 1: Listing the given quantities

Step 2: Understanding the concept of gravitational force and kinetic energy

Step 3: (a) Calculation of kinetic energy of satellite

Step 4: (b) Calculation of Gravitational force when the orbit radius is increased

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