a satellite orbits earth mass=5.98x10^24 once every 8 hours. At what radius does the satellite orbit
To determine the radius at which the satellite orbits the Earth, we can use the following formula:
\(T = 2\pi\sqrt{\frac{r^3}{GM}}\)
Where:
T = Orbital period (hours)
r = Radius of orbit (meters)
G = Gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
M = Mass of the Earth (approximately 5.98 × 10^24 kg)
Since the given orbital period is 8 hours, we can plug in the values into the formula and solve for r.
\(8 = 2\pi\sqrt{\frac{r^3}{6.67430 \times 10^{-11} \times 5.98 \times 10^{24}}}\)
Let's simplify the equation:
\(4 = \pi\sqrt{\frac{r^3}{2.0988 \times 10^{14}}}\)
Squaring both sides:
\(16 = \pi^2 \frac{r^3}{2.0988 \times 10^{14}}\)
Rearranging the equation to isolate \(r^3\):
\(r^3 = \frac{16 \times 2.0988 \times 10^{14}}{\pi^2}\)
Now, we can calculate the cube root of both sides to find the value of r:
\(r = \sqrt[3]{\frac{16 \times 2.0988 \times 10^{14}}{\pi^2}}\)
Evaluating this expression using a calculator, we find that the radius of the satellite's orbit around the Earth is approximately 2.67 × 10^7 meters.
Well, you know that
T = 2π√(r^3/GM)
So just look up G and plug in your numbers to find r.