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Q10.

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Found in: Page 132

Physics Principles with Applications

Book edition 7th
Author(s) Douglas C. Giancoli
Pages 978 pages
ISBN 978-0321625922

Two satellites orbit the Earth in circular orbits of the same radius. One satellite is twice as massive as the other. Which statement is true about the speeds of these satellites?(a) The heavier satellite moves twice as fast as the lighter one.(b) The two satellites have the same speed.(c) The lighter satellite moves twice as fast as the heavier one.(d) The ratio of their speeds depends on the orbital radius.

The correct option is (b) the two satellite have same speed.

See the step by step solution

Step 1. Understanding the orbital speed of the satellites

The orbital speed can be described as the speed with which satellites can move in a particular orbit. Orbital speed is dependent on the universal gravitational constant, mass of the Earth and the orbit radius.

Step 2. Expression of the orbital velocity

The orbital velocity of a satellite is given by the expression,

$v=\sqrt{\frac{G{M}_{\mathrm{e}}}{r}}$

Here, $G$ is the universal gravitational constant, ${M}_{\mathrm{e}}$ is the mass of the Earth and $r$ is the orbit radius.

The orbital velocity of a satellite is independent of the mass of the satellites. Since both satellites have equal radial distance from Earth’s surface, both move with the same orbital speed.

Step 3. Determination of the ratio of the orbital speeds of the satellites

The ratios of the orbital speeds can be expressed as,

$\begin{array}{c}\frac{{v}_{1}}{{v}_{2}}=\frac{\sqrt{\frac{G{M}_{\mathrm{e}}}{{r}_{1}}}}{\sqrt{\frac{G{M}_{\mathrm{e}}}{{r}_{2}}}}\\ =\sqrt{\frac{{r}_{2}}{{r}_{1}}}\\ =\sqrt{\frac{{r}_{1}}{{r}_{1}}}\\ =1\end{array}$

Here, ${r}_{1}$ is the radius of satellite (1), ${r}_{2}$ is the radius of satellite (2), ${v}_{1}$ is the orbital speed of satellite (1), ${v}_{2}$ is the orbital speed of satellite (2).

So, the ratio becomes one, and the orbital velocity of both satellites is equal.

Thus, option (d) is not correct.

Step 4. Determination of the acceleration of the object

The acceleration of the object can be expressed as,

${a}_{\mathrm{g}}=\frac{{F}_{\mathrm{g}}}{m}$

Here, ${F}_{\mathrm{g}}$ is the force due to gravity, m is the mass of the object.

Since all the objects (like both satellites) have the same radial distance from the surface of the Earth so, they are accelerated with the same centripetal acceleration.

Thus, from the above analysis, option (b) is correct.