The Canadian microsatellite MOST (Microvariability and Oscillations of Stars) has a mass of just 52 kg. It travels in an almost circular orbit at an average altitude of 820 km above Earth’s surface.
a) Calculate the gravitational force between Earth and the MOST satellite at this altitude.
b) What speed does the MOST satellite need to maintain its altitude? Express the speed in metres per second and kilometres per hour.
c) Determine the orbital Period of MOST.
(a) F=G•m•M/(R+h)²,
where the gravitational constant G =6.67•10^-11,
Earth’s mass is M = 5.97•10^24 kg,
Earth’s radius is R = 6.378•10^6 m.
h =8.2•10^5 m,
m = 52 kg.
(b) m•a = m•v²/(R+h) = G•m•M/(R+h)².
v =sqrt [G•M/(R+h)]
T =2•π• (R+h)/v =
=2•π•(R+h)/sqrt[G•M/(R+h].
a) To calculate the gravitational force between Earth and the MOST satellite, we can use Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant (approximately 6.67430 × 10^-11 N m^2/kg^2), m1 and m2 are the masses of the two objects (Earth and the satellite), and r is the distance between their centers of mass.
Let's assume the mass of the Earth is 5.972 × 10^24 kg (standard value), and the mass of the satellite is 52 kg.
The average altitude of the satellite is 820 km above Earth's surface. However, to find the distance between the centers of mass, we need to add the radius of the Earth to the altitude. The radius of the Earth is approximately 6,371 km.
So, the distance (r) between the centers of mass is (820 km + 6,371 km).
Now we can substitute these values into the formula:
F = (6.67430 × 10^-11 N m^2/kg^2) * (5.972 × 10^24 kg) * (52 kg) / ((820 km + 6,371 km)^2)
Calculate this expression to find the gravitational force between Earth and the satellite at this altitude.
b) To maintain its altitude, the MOST satellite needs to have the right speed to balance the gravitational force pulling it towards Earth and the centripetal force keeping it in orbit.
The gravitational force between Earth and the satellite provides the centripetal force:
F = m * (v^2) / r
where F is the gravitational force, m is the mass of the satellite, v is the velocity of the satellite, and r is the distance between the centers of mass.
Rearranging the equation, we can solve for the velocity v:
v = sqrt((F * r) / m)
Substitute the known values into the equation:
v = sqrt((gravitational force * r) / m)
Calculate this expression to find the speed of the satellite in meters per second. Then convert this to kilometers per hour by multiplying it by 3600 (since there are 3600 seconds in an hour).
c) The orbital period (T) of the satellite can be calculated using the formula:
T = 2 * pi * sqrt(r^3 / (G * m1))
where T is the orbital period, pi is a mathematical constant approximately equal to 3.14, r is the distance between the centers of mass, G is the gravitational constant, and m1 is the mass of the Earth.
Substitute the known values into the equation and calculate to find the orbital period of the satellite.