A spacecraft is in orbit around a planet. The radius of the orbit is 3.2 times the radius of the planet (which is R = 71490 km). The gravitational field at the surface of the planet is 24 N/kg. What is the period of the spacecraft's orbit?
To find the period of the spacecraft's orbit, we can use Kepler's Third Law, which relates the period of an orbit (T) to the radius of the orbit (r) and the mass of the planet (M).
Kepler's Third Law states that the square of the period of an orbit (T) is directly proportional to the cube of the radius of the orbit (r): T^2 ∝ r^3.
First, let's find the radius of the spacecraft's orbit (r):
Given that the radius of the spacecraft's orbit is 3.2 times the radius of the planet (R = 71490 km), we have: r = 3.2 * R.
Substituting the value of R into the equation, we get: r = 3.2 * 71490 km.
Now, let's convert the radius to meters (since the units of the gravitational field constant are in SI units):
1 km = 1000 m.
So, the radius of the orbit in meters is: r = 3.2 * 71490 km * 1000 m/km.
Next, we need to find the mass of the planet (M).
The gravitational field at the surface of the planet is given as 24 N/kg. This means that the acceleration due to gravity at the surface of the planet (g) is 24 m/s^2.
The gravitational field is defined as the gravitational force per unit mass, so we can use it to find the mass of the planet:
g = G * M / R^2,
where G is the gravitational constant (approximately 6.674 × 10^-11 N*m^2/kg^2) and R is the radius of the planet (71490 km).
Rearranging the equation, we can solve for the mass of the planet (M):
M = g * R^2 / G.
Substituting the given values, we get: M = 24 N/kg * (71490 km * 1000 m/km)^2 / (6.674 × 10^-11 N*m^2/kg^2).
Now that we have the radius of the spacecraft's orbit (r) and the mass of the planet (M), we can use Kepler's Third Law to find the period of the orbit (T):
T^2 = (r^3 * (4 * π^2)) / (G * M).
Substituting the values, we get: T^2 = ((3.2 * 71490 km * 1000 m/km)^3 * (4 * π^2)) / (6.674 × 10^-11 N*m^2/kg^2).
Taking the square root of both sides, we get: T = √(((3.2 * 71490 km * 1000 m/km)^3 * (4 * π^2)) / (6.674 × 10^-11 N*m^2/kg^2)).
Evaluating this equation will give us the period (T) of the spacecraft's orbit.