Find the orbital radius of a geosynchronous satellite. Do not assume the speed found in Example \(5.9 .\) Start by writing an equation that relates the period, radius, and speed of the orbiting satellite. Then apply Newton's second law to the satellite. You will have two equations with two unknowns (the speed and radius). Eliminate the speed algebraically and solve for the radius.

We can find the equation that relates the period (T), radius (r), and orbital speed (v) of the satellite with the following formula: \[v = \frac{2 \pi r}{T}\] This equation comes from the fact that the circumference of the orbit is \(2 \pi r\), and the satellite travels this distance in a time period T.

Next, we need to apply Newton's second law. The gravitational force acting on the satellite is the centripetal force that maintains its circular orbit: \[F_{gravity} = F_{centripetal}\] The gravitational force can be represented as: \[F_{gravity} = G \frac{M_{earth}m_{satellite}}{r^2}\] where \(G\) is the gravitational constant, \(M_{earth}\) is the mass of the Earth, and \(m_{satellite}\) is the mass of the satellite. The centripetal force can be represented as: \[F_{centripetal} = m_{satellite}\frac{v^2}{r}\] Now, we can equate the two forces: \[G \frac{M_{earth}m_{satellite}}{r^2} = m_{satellite}\frac{v^2}{r}\]

We need to eliminate the speed (v) from the equations. First, solve the equation from Step 1 for v: \[v = \frac{2 \pi r}{T}\] Now, square the equation: \[v^2 = \frac{4 \pi^2 r^2}{T^2}\] Insert this expression for \(v^2\) into the equation from Step 2: \[G \frac{M_{earth}m_{satellite}}{r^2} = m_{satellite}\frac{4 \pi^2 r^2}{T^2 r}\] We can cancel the \(m_{satellite}\) from both sides: \[G \frac{M_{earth}}{r^2} = \frac{4 \pi^2 r^2}{T^2 r}\]

Now, solve the resulting equation for the radius (r): \[r^3 = \frac{G M_{earth} T^2}{4 \pi^2}\] Taking the cube root of both sides: \[r = \sqrt[3]{\frac{G M_{earth} T^2}{4 \pi^2}}\] Since the satellite is geosynchronous, its period (T) is equal to the Earth's rotational period, 24 hours, or 86400 seconds. Plug in the known values: \[r = \sqrt[3]{\frac{(6.674 \times 10^{-11} \text{ m}^3\text{ kg}^{-1}\text{ s}^{-2})(5.972 \times 10^{24} \text{ kg})(86400\text{ s})^2}{4 \pi^2}}\] Finally, calculate the value of r: \[r \approx 42,164 \text{ km}\] This is the orbital radius of a geosynchronous satellite. Note that this value includes the radius of the Earth; to find the altitude above the Earth's surface, subtract the Earth's radius: \[Altitude \approx 42,164 \text{ km} - 6,371 \text{ km} = 35,793 \text{ km}\]