A satellite in circular orbit around the earth has an orbital speed of 5 km/s. What is the period of the orbit? Assume the mass of the Earth is 5.98e24 kg.
To find the period of the orbit, we can use the formula for the period of a circular orbit:
T = 2πr / v
Where T is the period, r is the radius of the orbit, and v is the orbital speed.
The radius of the orbit can be found using the formula for the gravitational force between the satellite and the Earth:
F = G * (m1 * m2) / r^2
Where F is the gravitational force, G is the gravitational constant (approximately 6.67430 x 10^-11 m^3 kg^-1 s^-2), m1 is the mass of the satellite, m2 is the mass of the Earth, and r is the radius of the orbit.
Solving this equation for r gives us:
r = √(G * m2 / F)
Since the satellite is in a circular orbit, the gravitational force is equal to the centripetal force:
F = m1 * v^2 / r
Simplifying this equation gives us:
r = m1 * v^2 / (G * m2)
Now we can substitute this value of r into the formula for the period:
T = 2π * (m1 * v^2 / (G * m2)) / v
Simplifying further gives us:
T = 2π * (m1 * v) / (G * m2)
Substituting the given values:
m1 = mass of the satellite = unknown
v = orbital speed = 5 km/s = 5000 m/s
m2 = mass of the Earth = 5.98 x 10^24 kg
G = gravitational constant = 6.67430 x 10^-11 m^3 kg^-1 s^-2
Now, solving for T:
T = 2π * (m1 * 5000) / (6.67430 x 10^-11 * 5.98 x 10^24)
To find the value of T, we need to know the mass of the satellite (m1).