i keep trying to put this in my calculator but i keep getting the wrong answer...can someone please try it on their calculator and help me?
thnks
(a) Find the distance r to geosynchronous orbit.
Apply Kepler's third law. T^2=((4pi^2)/(GM_E))r^3
Substitute the period in seconds, T = 86,400 s, the gravity constant G = 6.67 multiplied by 10-11 kg-1 m3/s2, and the mass of the Earth, ME = 5.98 multiplied by 1024 kg. Solve for r.
T^2=((4pi^2)/(GM_E))r^3
What equation are you using? You should be using Kepler's third law, and setting the satellite period to 23 hours 56 minutes (1 sidereal day)
Or check the derivation here:
http://en.wikipedia.org/wiki/Geostationary
To find the distance r to geosynchronous orbit using Kepler's third law, you can follow these steps:
Step 1: Determine the values of the variables given in the problem:
- Period in seconds (T) = 86,400 s
- Gravity constant (G) = 6.67 × 10^-11 kg^-1 m^3/s^2
- Mass of Earth (M_E) = 5.98 × 10^24 kg
Step 2: Substitute the given values into the equation:
T^2 = ((4π^2)/(GM_E))r^3
Substituting the values:
(86,400)^2 = ((4π^2)/(6.67 × 10^-11 × 5.98 × 10^24))r^3
Step 3: Simplify the equation:
86,400^2 = (4π^2 × 1.496 × 10^10 × 1.08 × 10^-34)r^3
Here, 1.496 × 10^10 is approximately the square of the average distance of the Earth from the Sun (known as the astronomical unit or AU) and 1.08 × 10^-34 is the product of the gravitational constant and the mass of the Earth.
Step 4: Solve for r by rearranging the equation:
r^3 = (86,400^2) / (4π^2 × 1.496 × 10^10 × 1.08 × 10^-34)
Compute the right-hand side of the equation:
r^3 ≈ 5.06 × 10^9 meters^3 / 8.94 × 10^44
Divide both sides by the constant on the right-hand side:
r^3 ≈ 5.663 × 10^-36
Take the cube root on both sides to solve for r:
r ≈ 1.720 × 10^-12 meters
So, the distance to geosynchronous orbit (r) is approximately 1.720 × 10^-12 meters.