Two artificial satellites, 1 and 2, orbit the Earth in circular orbits having radii R1 and R2, respectively, as shown above. If R2 = 2R1, the accelerations a2 and a1 of the two satellites are related by which of the following?
1)a2 = 4a1
2)a2 = 2a1
3)a2 = a1/4
4)a2 = a1
5)a2 = a1/2
The acceleration of a satellite in a circular orbit is given by the equation:
a = v^2 / r
where v is the velocity of the satellite and r is the radius of the orbit.
Since both satellites are in circular orbits, their velocities can be found using the formula:
v = 2πr / T
where T is the period of the orbit.
The period of an orbit is given by:
T = 2πr / v
where v is the velocity of the satellite.
Substituting the expression for v into the formula for T gives:
T = 2πr / (2πr / T) = T
This means that the period of both satellites is the same.
Given that R2 = 2R1, the radius of the second satellite is twice the radius of the first satellite.
Therefore, the velocity of the second satellite is half the velocity of the first satellite:
v2 = 2πR2 / T = 2 * (2πR1) / T = 4πR1 / T = 2 * (2πR1 / T) = 2v1
Using the formula for acceleration, we can now compare the accelerations of the two satellites:
a2 = v2^2 / R2 = (2v1)^2 / (2R1) = 4v1^2 / (2R1) = 2v1^2 / R1 = 2(a1)
Therefore, the relationship between a2 and a1 is given by:
a2 = 2a1
Therefore, the correct answer is option 2: a2 = 2a1.