Two artificial satellites, 1 and 2, orbit the Earth in circular orbits having radii R1 and R2, respectively. If R2 = 2R1, why are the accelerations A2 and A1 of the two satellites related with the equation A2= (A1)/4
The acceleration of an object in circular motion is given by the equation A = (V^2)/r, where V is the velocity of the object and r is its radius.
In this case, since both satellites are in circular orbits around the Earth, their accelerations can be related using this equation.
For satellite 1 with radius R1, its acceleration can be written as A1 = (V1^2)/R1.
For satellite 2 with radius R2 = 2R1, its acceleration can be written as A2 = (V2^2)/(2R1).
To relate A2 to A1, we need to find the relationship between V2 and V1.
Since the satellites are in circular orbits, the velocity of an object in circular motion can be calculated as V = 2πr/T, where T is the period of the orbit.
For satellite 1, the velocity V1 can be written as V1 = 2πR1/T1.
For satellite 2, the velocity V2 can be written as V2 = 2πR2/T2.
Since R2 = 2R1, the velocity V2 can also be written as V2 = 2π(2R1)/T2 = 4πR1/T2.
Now we can substitute these expressions for V1 and V2 back into the equations for A1 and A2.
A1 = (V1^2)/R1 = [(2πR1/T1)^2]/R1 = 4π^2R1^2/T1^2R1 = 4π^2R1/T1^2.
A2 = (V2^2)/(2R1) = [(4πR1/T2)^2]/(2R1) = 16π^2R1^2/(2T2^2R1) = 8π^2R1/T2^2.
Dividing A2 by A1 gives us:
A2/A1 = (8π^2R1/T2^2)/(4π^2R1/T1^2) = (8π^2R1/T2^2)*(T1^2/4π^2R1) = T1^2/T2^2.
Since the period of an orbit is proportional to the square root of the radius (T ∝ √r), we have T1^2/T2^2 = (R1/R2)^2 = (1/2)^2 = 1/4.
Therefore, A2/A1 = 1/4 or A2 = (A1)/4.