Speed of an Earth satellite in a circular orbit (10-11)

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Question

Speed of an Earth satellite in a circular orbit
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Review

We can find the speed \(v\) required for a circular orbit of radius \(r\) from Equation 3-17 for the acceleration in uniform circular motion \(a_{\mathrm{cent}} = v^2/r\). For a satelite of mass \(m\) orbiting Earth, the acceleration is provided by Earth's gravitational force. From Newton's second law, this says

\(F_{\mathrm{Earth\ on\ satelite}} = ma_{\mathrm{cent}}\)

or from Equation 3-17 for uniform circular motion and Equation 10-2 for the law of universal gravitation,

\(\frac{Gm_{\mathrm{Earth}}m}{r^2} = m\frac{v^2}{r}\)

To solve for \(v\), multiply both sides of this equation by \(r/m\)

\(\frac{Gm_{\mathrm{Earth}}}{r} = v^2\)

and take the square root: