A satellite orbits the moon at height of 2000m.assuming it to gravitational pull of the moon,find the speed and time it takes for one orbit,given that mass of moon is 7.34 * 10 to power 22 kg and radius is 1.738 * 10 to power 6m.(g = 6.6 * 10 to power -11nm2/kg).
remember the orbital radisu is rmoon+altitude.
Force=GMm/orbitRadius^2
and
force= m v^2/orbitRAdius
set them equal, solve for
v.
then, period T:
v=2PI*orbitalradius/T
solve for T
To find the speed and time it takes for one orbit, we can use the law of gravitation and the centripetal acceleration formula.
Step 1: Find the gravitational force between the satellite and the moon using the law of gravitation:
Force = (G * m1 * m2) / r^2
Where:
G is the gravitational constant (6.6 * 10^-11 Nm^2/kg^2)
m1 is the mass of the satellite
m2 is the mass of the moon
r is the distance between the satellite and the moon
Given:
Height of the satellite from the moon's surface (h) = 2000 m
Radius of the moon (R) = 1.738 * 10^6 m
Mass of the moon (M) = 7.34 * 10^22 kg
The radius between the satellite and the center of the moon is the sum of the height of the satellite and the radius of the moon:
r = R + h = (1.738 * 10^6 m) + (2000 m) = 1.740 * 10^6 m
Substituting the values into the equation:
Force = (6.6 * 10^-11 Nm^2/kg^2) * (m1 * 7.34 * 10^22 kg) / (1.740 * 10^6 m)^2
Step 2: Equate the gravitational force with the centripetal force:
Centripetal Force = m1 * (v^2 / r)
Where:
m1 is the mass of the satellite
v is the velocity of the satellite
r is the distance between the satellite and the center of the moon
Setting the gravitational force equal to the centripetal force:
(G * m1 * m2) / r^2 = m1 * (v^2 / r)
Canceling out the mass of the satellite, m1:
(G * m2) / r = v^2 / r
Step 3: Solve for the velocity (v):
v^2 = (G * m2) / r
v = √((G * m2) / r)
Now we can calculate the speed of the satellite.
Substituting the values:
v = √((6.6 * 10^-11 Nm^2/kg^2 * 7.34 * 10^22 kg) / (1.740 * 10^6 m))
v ≈ √(3.526 * 10^12 Nm^2/kg^2) ≈ 1.877 * 10^6 m/s
Therefore, the speed of the satellite is approximately 1.877 * 10^6 m/s.
Step 4: To find the time it takes for one orbit, we divide the circumference of the orbit by the speed of the satellite.
The circumference of the orbit is:
Circumference = 2π * r
Substituting the values:
Circumference = 2π * (1.740 * 10^6 m)
The time it takes for one orbit is:
Time = Circumference / Speed
Substituting the values:
Time = (2π * (1.740 * 10^6 m)) / (1.877 * 10^6 m/s)
Time ≈ 5.857 seconds
Therefore, the time it takes for one orbit is approximately 5.857 seconds.