A satellite has a mass of 6146 kg and is in a circular orbit 4.57 × 105 m above the surface of a planet. The period of the orbit is 1.6 hours. The radius of the planet is 4.44 × 106 m. What would be the true weight of the satellite if it were at rest on the planet’s surface?
What do I use to find the mass of the unknown planet?
The final equation to use is: W=(GmM)/r^2
m=mass of satelite
M=mass of planet
What is the equation to find the mass of the planet?
The weight of the planet in orbit is
W = M V^2/R
where R = 4.57*10^5 m and V is the velocity at that altitude,
V = 2 pi R/(period).
Calculate W.
The weight W' of the satellite at the planet's surface is
W'/W = (4.44*10^6/4.57*10^5)^2
because weight is inversely proportional to the square of distance.
The number you want is W'
To find the mass of the unknown planet, you can use the law of universal gravitation and the given information about the satellite's orbit. The formula for the period of an orbit is given by:
T = 2π √(r^3 / (G*M))
Where T is the period, r is the distance between the satellite and the center of the planet, G is the gravitational constant, and M is the mass of the planet.
First, let's rearrange the formula to solve for M:
M = (4π^2 * r^3) / (G * T^2)
Given values:
- r = 4.44 × 10^6 m (radius of the planet)
- T = 1.6 hours = 1.6 * 60 * 60 = 5,760 seconds (period of the orbit)
- G = 6.67430 × 10^-11 m^3 kg^-1 s^-2 (gravitational constant)
Now we can substitute these values into the formula to find the mass of the unknown planet:
M = (4π^2 * (4.44 × 10^6)^3) / (6.67430 × 10^-11 * (5,760)^2)
Calculating this equation will give you the mass of the unknown planet in kilograms.