4(a) What is the magnitude of the gravitational force of the Earth on a surveillance
satellite of mass 1040 kg that is in a grazing circular orbit 100 km above the
surface of the Earth.
(b) Find the magnitude of the acceleration of a satellite in such a grazing orbit.
(c) Determine the speed of the satellite.
Take the mass and radius of the Earth as 5.98 × 1024 kg and 6.37 × 106 m
respectively.
The gravitational constant
G =6.67•10⁻¹¹ N•m²/kg²,
h=100000 m
m=1040 kg
Earth’s mass is M = 5.98•10²⁴kg,
Earth’s radius is R = 6.37 •10⁶ m.
(a) F =G•m •M/(R+h)²
(b) a= G• M/(R+h)²
(c) mv²/(R+h) = G•m •M/(R+h)²
v=sqrt (G•M/(R+h))
To answer these questions, we will use the equation for the gravitational force between two objects:
F = (G * m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant (approximately 6.674 × 10^-11 N m^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between the centers of the two objects.
(a) To find the magnitude of the gravitational force of the Earth on the surveillance satellite, we need to know the mass of the Earth and the mass of the satellite. Given that the mass of the Earth (m1) is 5.98 × 10^24 kg and the mass of the satellite (m2) is 1040 kg, we can substitute these values into the equation to calculate the force.
F = (G * m1 * m2) / r^2
F = (6.674 × 10^-11 N m^2/kg^2) * (5.98 × 10^24 kg) * (1040 kg) / (100,000 m + radius of the Earth)^2
To find the radius of the Earth plus the distance of the satellite above the surface of the Earth, we add the given values:
radius of the Earth + 100,000 m = (6.37 × 10^6 m + 100,000 m)
Once we calculate this sum, we substitute it back into the equation to find the magnitude of the gravitational force.
(b) To find the magnitude of the acceleration of the satellite in the grazing orbit, we need to equate the gravitational force with the centripetal force. The centripetal force is given by the formula:
F = m * a_c
where F is the gravitational force, m is the mass of the satellite, and a_c is the centripetal acceleration.
Using the gravitational force calculated in part (a), we equate it with the centripetal force:
F = m * a_c
Solve for a_c:
a_c = F / m
Substitute the gravitational force and the mass of the satellite into this equation to find the magnitude of the acceleration.
(c) To find the speed of the satellite, we use the formula for the centripetal acceleration:
a_c = v^2 / r
where v is the speed of the satellite and r is the distance from the center of the Earth to the satellite (radius of the Earth + height above the surface). We know the radius of the Earth and the height above the surface, so we can substitute these values into the equation to solve for the speed.
These calculations will give us the answers to the questions. Just remember to substitute the provided values into the equations and perform the necessary calculations.