A satellite has a mass of 6000 kg and is in a circular orbit 3.30 105 m above the surface of a planet. The period of the orbit is two hours. The radius of the planet is 4.20 106 m. What is the true weight of the satellite when it is at rest on the planet's surface?
I know I'm supposed to use g=Gm1m2/R^2 and r=Gm1/v^2 but I keep getting it wrong.
your g formula is most wrong.
centripetal force= gravitationalforce
m*v^2/r=G*Me*m/(radplanet+ heightabove)^2
now, note you do not need the mass of the satellite, however, you need the mass of the Planet.
velociy= 2PI (radplanet+heightabove)/period
To find the true weight of the satellite when it is at rest on the planet's surface, we need to first calculate the mass of the planet.
Given:
Mass of the satellite (m2) = 6000 kg
Height of the satellite's orbit (h) = 3.30 × 10^5 m
Radius of the planet (R) = 4.20 × 10^6 m
Period of the orbit (T) = 2 hours = 2 × 60 × 60 = 7200 seconds
To calculate the mass of the planet, we can use the formula:
GMm2 / R^2 = (m1 v^2) / r
Where:
G is the gravitational constant (6.674 × 10^-11 Nm^2/kg^2)
M is the mass of the planet (which we want to find)
m2 is the mass of the satellite
R is the distance from the center of the planet to the satellite's orbit
Rearranging the formula to solve for M:
M = (m2 R^2 v^2) / (G r)
Now we can substitute the known values:
M = (6000 kg × (4.20 × 10^6 m)^2 × v^2) / (6.674 × 10^-11 Nm^2/kg^2 × 3.30 × 10^5 m)
Next, we need to calculate the velocity (v) of the satellite in its orbit. Since the satellite is in a circular orbit, we can use the formula for centripetal acceleration:
a = v^2 / r
Where:
a is the acceleration
v is the velocity
r is the distance from the center of the planet to the satellite's orbit
Since the satellite is at rest on the planet's surface, the acceleration is equal to the acceleration due to gravity (g) on the planet's surface.
So we have:
g = v^2 / r
Rearranging the formula to solve for v:
v = sqrt(g × r)
Substituting known values:
v = sqrt((g × 4.20 × 10^6 m)
Next, we can substitute the calculated value of v back into the formula for M:
M = (6000 kg × (4.20 × 10^6 m)^2 × (sqrt(g × r))^2) / (6.674 × 10^-11 Nm^2/kg^2 × 3.30 × 10^5 m)
Now we can substitute the value of g:
The acceleration due to gravity (g) can be calculated using the formula:
g = G × (M / R^2)
Rearranging the formula to solve for g:
g = (G × M) / R^2
Substituting known values:
g = (6.674 × 10^-11 Nm^2/kg^2 × M) / (4.20 × 10^6 m)^2
Now we can substitute this expression back into the formula for M:
M = (6000 kg × (4.20 × 10^6 m)^2 × (sqrt((6.674 × 10^-11 Nm^2/kg^2 × M) / (4.20 × 10^6 m)^2) × 3.30 × 10^5 m))^2) / (6.674 × 10^-11 Nm^2/kg^2 × 3.30 × 10^5 m)
This equation appears to be nonlinear and cannot be solved directly. However, we can use numerical methods or approximation techniques to solve it.