A new moon is discovered orbiting Neptune with an orbital speed of 9.3 ´ 10 m/s. Neptune’s mass is 1.0 ´ 10 kg. What is the radius of the new moon’s orbit? What is the orbital period? Assume that the orbit is circular. (G = 6.673 ´ 10 N·m/kg)
To find the radius of the new moon's orbit, we can use the formula for centripetal force:
F = (mv^2) / r
Where F is the gravitational force between the moon and Neptune, m is the mass of the moon, v is the orbital speed, and r is the radius of the orbit.
The gravitational force between the moon and Neptune can be calculated using the equation:
F = (G * m1 * m2) / r^2
Where G is the gravitational constant (6.673 × 10^-11 N·m^2/kg^2), m1 is the mass of Neptune, m2 is the mass of the moon, and r is the distance between the centers of the two bodies.
Since we know the orbital speed (v = 9.3 × 10^3 m/s) and the mass of Neptune (m1 = 1.0 × 10^26 kg), we can rearrange the equations and solve for the radius (r):
F = (mv^2) / r
r = (mv^2) / F
F = (G * m1 * m2) / r^2
r^3 = (G * m1 * m2) / (v^2)
r = (G * m1 * m2)^(1/3) / v * 2/3
Substituting the given values:
r = (6.673 × 10^-11 N·m^2/kg^2 * 1.0 × 10^26 kg * 1.0 × 10^24 kg)^(1/3) / (9.3 × 10^3 m/s) * 2/3
Calculating the value:
r ≈ 2.61 × 10^8 meters
So, the radius of the new moon's orbit is approximately 2.61 × 10^8 meters.
To find the orbital period, we can use Kepler's third law of planetary motion, which states:
T^2 = (4π^2 * r^3) / (G * m1)
Where T is the orbital period, r is the radius of the orbit, G is the gravitational constant, and m1 is the mass of Neptune.
Rearranging the equation to solve for T:
T = √((4π^2 * r^3) / (G * m1))
Substituting the given values:
T = √((4π^2 * (2.61 × 10^8)^3) / (6.673 × 10^-11 N·m^2/kg^2 * 1.0 × 10^26 kg))
Calculating the value:
T ≈ 5.56 × 10^6 seconds
So, the orbital period of the new moon is approximately 5.56 × 10^6 seconds.
To find the radius of the new moon's orbit, we need to use the formula for centripetal force and equate it to the gravitational force between the moon and Neptune.
The centripetal force is given by:
Fc = m*v^2 / r
Where Fc is the centripetal force, m is the mass of the moon, v is the orbital speed, and r is the radius of the orbit.
The gravitational force between the moon and Neptune is given by:
Fg = G*m*M / r^2
Where Fg is the gravitational force, G is the gravitational constant, m is the mass of the moon, M is the mass of Neptune, and r is the radius of the orbit.
Since the two forces are equal, we can equate them:
m*v^2 / r = G*m*M / r^2
We can cancel out the mass of the moon:
v^2 / r = G*M / r^2
Multiplying both sides by r:
v^2 = G*M / r
Now we can solve for the radius of the orbit:
r = G*M / v^2
Substituting the given values:
r = (6.673 ´ 10^-11 N·m^2 / kg^2) * (1.0 ´ 10^26 kg) / (9.3 ´ 10^3 m/s)^2
Calculating the above expression, we find:
r ≈ 2.00 ´ 10^12 meters
To find the orbital period, we can use the formula for the period of a circular orbit:
T = 2π * r / v
Where T is the orbital period, r is the radius of the orbit, and v is the orbital speed.
Substituting the given values:
T = 2π * (2.00 ´ 10^12 meters) / (9.3 ´ 10^3 m/s)
Calculating the above expression, we find:
T ≈ 1.71 ´ 10^8 seconds
Therefore, the radius of the new moon's orbit is approximately 2.00 ´ 10^12 meters, and the orbital period is approximately 1.71 ´ 10^8 seconds.