Space explorers discover an 8.7×10^17kg asteroid that happens to have a positive charge of 4400
C. They would like to place their 3.3×10^5kg spaceship in orbit around the asteroid. Interestingly, the solar wind has given their spaceship a charge of -1.2C. What speed must their spaceship have to achieve a 7300-
km -diameter circular orbit?
I do not know how to start this problem.
6.2 m/s
To solve this problem, we can use the concept of electrostatic force and centripetal force.
Step 1: Calculate the gravitational force between the spaceship and the asteroid using Newton’s law of universal gravitation:
F_gravity = (G * m1 * m2) / r^2
Where:
- F_gravity is the gravitational force between the spaceship and the asteroid,
- G is the gravitational constant (6.67 × 10^-11 N m^2/kg^2),
- m1 is the mass of the spaceship (3.3 × 10^5 kg),
- m2 is the mass of the asteroid (8.7 × 10^17 kg),
- r is the radius of the circular orbit (half of the diameter, which is 7300 km / 2 = 3650 km = 3.65 × 10^6 m).
Step 2: Calculate the electrostatic force between the spaceship and the asteroid using Coulomb's law:
F_electrostatic = k * |q1 * q2| / r^2
Where:
- F_electrostatic is the electrostatic force between the spaceship and the asteroid,
- k is the electrostatic constant (9 × 10^9 N m^2/C^2),
- q1 is the charge of the spaceship (-1.2 C),
- q2 is the charge of the asteroid (4400 C),
- r is the radius of the circular orbit (3.65 × 10^6 m).
Step 3: Set the gravitational force equal to the electrostatic force, since both forces must balance each other for the spaceship to be in a stable orbit:
F_gravity = F_electrostatic
Step 4: Since the centripetal force required to keep the spaceship in a circular orbit is equal to the electrostatic force, we can solve for the speed:
F_centripetal = F_electrostatic
Using the relationship between centripetal force and speed:
F_centripetal = (m * v^2) / r
Where:
- F_centripetal is the centripetal force required to keep the spaceship in orbit,
- m is the mass of the spaceship (3.3 × 10^5 kg),
- v is the speed of the spaceship we want to find,
- r is the radius of the circular orbit (3.65 × 10^6 m).
Step 5: Substitute the gravitational force and electrostatic force expressions into the centripetal force equation:
(G * m1 * m2) / r^2 = k * |q1 * q2| / r^2
(m * v^2) / r = k * |q1 * q2| / r^2
Step 6: Rearrange the equation and solve for the speed of the spaceship:
v = √((G * m2 * q1 * q2) / k * m1)
m = spaceship mass = 3.3*10^5
M = asteroid mass = 8.7*10^17
R = 7,300,000 m = 7.3*10^6 meters
F = m a = m v^2/R
F = G m M/R^2 + k |Q1| |Q2|/R^2
(both forces +, attraction)
so
v^2 = G M/R + k |Q1| |Q2|/(m R)