Chapter 13: Q64P (page 383)

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 $3.0 \times 10^{7} m$ ?

Short Answer

Expert verified

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

Step by step solution

Listing the given quantities

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

$F_{1} = 80 N$

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

Understanding the concept of gravitational force and kinetic energy

Using the given gravitational force and kinetic energy formula, we can findthekinetic 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:

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

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

(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),

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

(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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Short Answer

Step by step solution

Listing the given quantities

Understanding the concept of gravitational force and kinetic energy

(a) Calculation of kinetic energy of satellite

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

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A satellite orbits a planet of unknown mass in a circle of radius 2.0×107m. 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 to3.0×107m ?

Short Answer

Step by step solution

Listing the given quantities

Understanding the concept of gravitational force and kinetic energy

(a) Calculation of kinetic energy of satellite

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Medical Physics

Circular Motion and Gravitation

Solid State Physics

Engineering Physics

Mechanics and Materials

Nuclear Physics

Study anywhere. Anytime. Across all devices.