A satellite is orbiting a distant planet of mass 9.6e25 kg and radius 86700 km. The satellite orbits at a height of 8610 km above the surface of the planet.
a) Determine the acceleration due to gravity at the location of the satellite orbit.
b) Determine the velocity of the satellite in this orbit.
c) What is the period of the satellite in this orbit?
weight of stationary object there = m g = G m M/r^2
so
g =
6.67*10^-11*9.6*10^25
/(86,700,000+8,610,000)^2
b) v^2/r = g
c) T = 2 pi r/v
To answer these questions, we need to use Newton's law of universal gravitation and the equations for centripetal acceleration, velocity, and period in circular motion.
a) The acceleration due to gravity at the location of the satellite orbit can be found using the formula:
a = G * (M / r^2)
Where a is the acceleration due to gravity, G is the gravitational constant (approximately 6.67430 × 10^(-11) N m^2 / kg^2), M is the mass of the planet, and r is the distance from the center of the planet to the satellite orbit.
Plugging in the values:
a = (6.67430 × 10^(-11) N m^2 / kg^2) * (9.6e25 kg) / (87710 km)^2
Note: We should convert the radius of the planet and the height of the satellite to meters (1 km = 1000 m) to get consistent units.
a = (6.67430 × 10^(-11) N m^2 / kg^2) * (9.6e25 kg) / (87710 * 10^3 m)^2
Now, we can calculate the value of acceleration due to gravity.
b) The velocity of the satellite in this orbit can be found using the formula:
v = √(G * M / r)
Where v is the velocity of the satellite, G is the gravitational constant, M is the mass of the planet, and r is the distance from the center of the planet to the satellite orbit.
Plugging in the values:
v = √((6.67430 × 10^(-11) N m^2 / kg^2) * (9.6e25 kg) / (87710 * 10^3 m))
c) The period of the satellite in this orbit can be found using the formula:
T = 2π * (r / v)
Where T is the period of the satellite, r is the distance from the center of the planet to the satellite orbit, and v is the velocity of the satellite.
Plugging in the values:
T = 2π * ((87710 * 10^3 m) / v)
Now, let's calculate the values for b) and c).