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:

Question

1) A2 = 4*A1
2) A2 = 2*A1
3) A1 = (A1)/4
4) A2= A1
5) A2= (A1)/2

Answer 1

The acceleration of an object in circular motion can be determined using the formula:

A = (v^2) / r

where A is the acceleration, v is the velocity, and r is the radius of the circular orbit.

Since both satellites are in circular motion, their accelerations can be determined using this formula.

For satellite 1:
A1 = (v1^2) / R1

For satellite 2:
A2 = (v2^2) / R2

We know that R2 = 2R1, so we can substitute this value into the equation for A2:
A2 = (v2^2) / (2R1)

To compare the accelerations A1 and A2, we need to find a relationship between v1 and v2.

The velocity of an object in circular motion can be determined using the formula:

v = 2πr / T

where v is the velocity, r is the radius of the circular orbit, and T is the period of the orbit.

We can use this formula to express the velocities v1 and v2 in terms of R1 and R2.

For satellite 1:
v1 = 2πR1 / T1

For satellite 2:
v2 = 2πR2 / T2

Since both satellites are orbiting the same Earth, the periods T1 and T2 will be the same. Therefore, we can substitute these values into the equations for v1 and v2:
v1 = 2πR1 / T
v2 = 2πR2 / T

We can now substitute these values of v1 and v2 into the equations for A1 and A2:

A1 = (v1^2) / R1
= [(2πR1 / T)^2] / R1
= (4π^2R1^2) / (T^2R1)
= 4π^2R1 / T^2

A2 = (v2^2) / (2R1)
= [(2πR2 / T)^2] / (2R1)
= (4π^2R2^2) / (2T^2R1)
= (2π^2R2^2) / (T^2R1)
= (2π^2(2R1)^2) / (T^2R1)
= (8π^2R1^2) / (T^2R1)
= 8π^2R1 / T^2

From these equations, we can see that A2 = 8π^2R1 / T^2 and A1 = 4π^2R1 / T^2.

Therefore, the relationship between A2 and A1 is:

A2 = 2 * A1

So, the correct answer is option 2: A2 = 2 * A1.