Two artificial satellites, 1 and 2, orbit the Earth in circular orbits having radii R1 and R2, respectively. If R2 = 2R1, how are the accelerations A2 and A1 of the two satellites related:
1) A2 = 4*A1
2) A2 = 2*A1
3) A2 = (A1)/4
4) A2= A1
5) A2= (A1)/2
The acceleration of an object in circular motion is given by the centripetal acceleration formula, which is:
A = (V^2) / R
where A is the acceleration, V is the velocity, and R is the radius of the circular orbit.
Since the satellites are in circular orbits, their velocities are related to their radii by the equation:
V = (2πR) / T
where T is the period of the orbit.
Combining these two equations, we can express the acceleration in terms of the radius:
A = (4π^2 R) / (T^2)
Since the period T is the same for both satellites (assuming they have the same orbital speed), we can compare their accelerations based on the ratio of their radii:
A2 / A1 = [(4π^2 R2) / (T^2)] / [(4π^2 R1) / (T^2)]
Simplifying this expression, we find:
A2 / A1 = R2 / R1
Given that R2 = 2R1, we can substitute this value into the equation and simplify further:
A2 / A1 = (2R1) / R1 = 2
Therefore, the correct answer is:
5) A2 = (A1) / 2