The physics of satellite motion around Jupiter.
A satellite of mass 2.00 x 104 kg is placed in orbit 6.00 x 105 m above the surface of
Jupiter. Please refer to the data table for planetary motion included in this lesson.
a) Determine the force of gravitational attraction between the satellite and Jupiter.
b) What must be the orbital speed of the satellite?
c) What must be the value of the gravitational field constant, g, at the location of the satellite?
d) One of the moons of Jupiter is Europa. Its period of motion is 3.07 x 10^5 s.
What must be the radius of its orbit?
e) If a satellite was placed in orbit around Europa at a height of 100.0 km above the surface of Europa, and period of motion was 7.58 x 10^3 s, what must be the mass of Europa? The radius of the moon
Europa is 1.57 x 10^6 m.
To answer these questions, we will use the following formulas:
1. Gravitational force between two objects:
F = G * (m1 * m2 / r^2)
2. Centripetal force in circular motion:
F_c = m * v^2 / r
3. Gravitational field strength:
g = G * M / r^2
Where:
- F is the gravitational force
- G is the gravitational constant ( 6.67 x 10^-11 N m^2/kg^2)
- m1 and m2 are the masses of the two objects
- r is the distance between the center of mass of the two objects
- F_c is the centripetal force
- m is the mass of the satellite/moon
- v is the orbital speed
- M is the mass of Jupiter/Europa
- g is the gravitational field constant
Now, let's calculate step by step:
a) Determine the force of gravitational attraction between the satellite and Jupiter.
Given:
Mass of the satellite (m1) = 2.00 x 10^4 kg
Mass of Jupiter (m2) = 1.90 x 10^27 kg
Distance (r) = 6.00 x 10^5 m
Using the formula, F = G * (m1 * m2 / r^2):
F = (6.67 x 10^-11 N m^2/kg^2) * (2.00 x 10^4 kg * 1.90 x 10^27 kg) / (6.00 x 10^5 m)^2
Calculate the force.
b) What must be the orbital speed of the satellite?
To find the orbital speed, we use the centripetal force formula:
F_c = m * v^2 / r
Rearranging the formula, v^2 = (F_c * r) / m
Plug in the known values:
F_c = The gravitational force between the satellite and Jupiter from part (a)
r = Distance from the center of Jupiter to the satellite's orbit
m = Mass of the satellite
Calculate the orbital speed.
c) What must be the value of the gravitational field constant, g, at the location of the satellite?
We use the gravitational field strength formula:
g = G * M / r^2
Given:
Mass of Jupiter (M) = 1.90 x 10^27 kg
Distance (r) = 6.00 x 10^5 m
Calculate the value of g.
d) One of the moons of Jupiter is Europa. Its period of motion is 3.07 x 10^5 s. What must be the radius of its orbit?
To find the radius of the orbit, we use Kepler's third law of planetary motion which states that the square of the orbital period (T) is proportional to the cube of the semi-major axis (r).
Given:
Period of motion (T) = 3.07 x 10^5 s
Calculate the radius of the orbit.
e) If a satellite was placed in orbit around Europa at a height of 100.0 km above the surface of Europa, and the period of motion was 7.58 x 10^3 s, what must be the mass of Europa? The radius of the moon Europa is 1.57 x 10^6 m.
Given:
Height of the satellite above Europa's surface = 100.0 km = 1.00 x 10^5 m
Period of motion (T) = 7.58 x 10^3 s
Radius of Europa (r) = 1.57 x 10^6 m
We can use the same formula from part (d) to find the mass of Europa.
Rearrange the formula to solve for M:
M = 4 * pi^2 * r^3 / G * T^2
Calculate the mass of Europa.
To answer these questions, we need to use the laws of planetary motion and gravitational force.
a) The force of gravitational attraction between two objects can be calculated using Newton's law of universal gravitation:
F = (G * m1 * m2) / r^2
Where F is the force of gravitational attraction, G is the gravitational field constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers of mass.
In this case, the satellite is orbiting Jupiter, so m1 is the mass of the satellite (2.00 x 10^4 kg), m2 is the mass of Jupiter (found in the data table), and r is the distance between the satellite and Jupiter's center (6.00 x 10^5 m + radius of Jupiter).
b) The orbital speed of a satellite can be calculated using the equation:
v = sqrt((G * M) / r)
Where v is the orbital speed, G is the gravitational field constant, M is the mass of the planet (found in the data table), and r is the distance between the satellite and the center of the planet.
c) The value of the gravitational field constant, G, is a fundamental constant in physics and has a fixed value of 6.67430 x 10^-11 m^3 kg^-1 s^-2.
d) The radius of the orbit of a moon can be determined using Kepler's third law of planetary motion:
T^2 = (4 * pi^2 * r^3) / (G * M)
Where T is the period of motion, r is the radius of the orbit, G is the gravitational field constant, and M is the mass of the planet.
In this case, we are given the period of motion (3.07 x 10^5 s) and need to solve for the radius of the orbit.
e) To calculate the mass of the moon Europa, we can rearrange the equation for the orbital speed:
v = sqrt((G * M) / (r + h))
Where v is the orbital speed, G is the gravitational field constant, M is the mass of Europa, r is the radius of Europa's moon, and h is the height of the satellite above the surface of Europa.
In this case, we are given the orbital speed (7.58 x 10^3 s) and the height above the surface of Europa (100.0 km), and need to solve for the mass of Europa.