Two satellites are orbiting the Earth in stable circular orbits. Satellite A has a mass m and is located at a distance 2r from the center of the Earth. Satellite B has a mass 2m and is located at a distance r from the center of the Earth. Which one of the following statements concerning this situation is false?
The mass that a satellite has does not affect its orbital speed
The gravitational force of the Earth acting on satellite B is greater than the Earth's gravitational force acting on satellite A.
The centripetal force on satellite B is eight times greater than the centripetal force on satellite A.
Satellite B has a greater orbital speed than satellite A does.
Satellite A has a greater centripetal acceleration than satellite B does.
To determine which statement is false, let's go through each statement and see if it is true or false.
Statement 1: The mass that a satellite has does not affect its orbital speed.
This statement is true. The mass of a satellite does not affect its orbital speed. Theoretically, if we assume other factors remain constant, two satellites of different masses can have the same orbital speed if they are at the same distance from the center of the Earth.
Statement 2: The gravitational force of the Earth acting on satellite B is greater than the Earth's gravitational force acting on satellite A.
This statement is true. The gravitational force acting on an object depends on the mass of the object and the distance from the center of the Earth. Since satellite B has a greater mass than satellite A, it will experience a greater gravitational force from the Earth.
Statement 3: The centripetal force on satellite B is eight times greater than the centripetal force on satellite A.
This statement is false. The centripetal force required to keep an object in circular motion depends on the mass of the object and the square of its orbital speed. Since satellite B has a mass of 2m and satellite A has a mass of m, the centripetal force on B would be four times greater than the centripetal force on A, not eight times.
Statement 4: Satellite B has a greater orbital speed than satellite A does.
This statement is true. The orbital speed of a satellite is determined by the gravitational force acting on it and the distance from the center of the Earth. Since satellite B is closer to the center of the Earth than satellite A, it will experience a greater gravitational force and therefore have a greater orbital speed.
Statement 5: Satellite A has a greater centripetal acceleration than satellite B does.
This statement is false. Centripetal acceleration depends on the square of the orbital speed and the radius of the orbit. Since satellite B has a smaller radius (r) than satellite A (2r), it will have a greater centripetal acceleration.
In conclusion, the false statement is statement 3: "The centripetal force on satellite B is eight times greater than the centripetal force on satellite A."
The false statement in this situation is: "Satellite A has a greater centripetal acceleration than satellite B does."
Explanation:
According to the equation for centripetal acceleration, which is given by ac = v^2/r (where ac is the centripetal acceleration, v is the orbital speed, and r is the radius of the orbit), the centripetal acceleration is directly proportional to the square of the orbital speed and inversely proportional to the radius of the orbit.
In this scenario, satellite A is at a distance 2r from the center of the Earth, while satellite B is at a distance r. Since the radius of satellite A's orbit is twice that of satellite B's orbit, the centripetal acceleration of satellite A will be one-fourth (1/2^2) of that of satellite B.
Therefore, the statement "Satellite A has a greater centripetal acceleration than satellite B does" is false.