Two artificial satellites, 1 and 2, orbit the Earth in circular orbits having radii R1 and R2. If R2=2R1, the accelerations a2 and a1 of the satellites are related how?
The centripetal acceleration of a satellite in a circular orbit is V^2/R , but V is a function of R also. For an earth satellite
GM/R^2 = V^2/R
where g is the universal constant of gravity and M is the mass of the earth. Therefore
V^2 = GM/R, and V^2 is inversely proportional to R.
That means that the acceleration V^2/R is inversely proportional to R^2. So if R2 = 2 R1, the acceleration at 2 is 1/4 as much.
I could have obtained this result more easily by recognizing that the acceleration is the gravitation force divided by satellite mass m, and the force is inversely proportional to R^2.
gmh
The accelerations of the satellites can be related using the formula for centripetal acceleration.
The centripetal acceleration of an object moving in a circle of radius R with a constant speed v is given by the equation:
a = v^2 / R
In this case, we can compare the accelerations of satellites 1 and 2 by using their respective radii R1 and R2. Given that R2 = 2R1, we can substitute this value into the equation:
a2 = v^2 / R2
a1 = v^2 / R1
Now, let's divide these two equations to compare their accelerations:
a2 / a1 = (v^2 / R2) / (v^2 / R1)
The v^2 terms cancel out:
a2 / a1 = R1 / R2
Since R2 = 2R1, we can substitute this into the equation:
a2 / a1 = R1 / (2R1)
Simplifying further:
a2 / a1 = 1 / 2
Therefore, the relationship between the accelerations of the satellites is a2 = 0.5 * a1. This means that the acceleration of satellite 2 is half of the acceleration of satellite 1.
To determine the relationship between the accelerations of the two satellites, we can use the formula for centripetal acceleration. Centripetal acceleration is given by the equation:
a = (v^2) / r
Where:
a = acceleration
v = linear velocity
r = radius of the orbit
In circular motion, the velocity v can be expressed as:
v = (2 * π * r) / T
Where:
T = period of revolution
Now, since the satellites are in circular orbits, we can assume that the periods of revolution are the same. Therefore, we can equate the velocities of the two satellites:
v2 = v1
Substituting the equation for velocity into the centripetal acceleration formula:
a2 = ((2 * π * r2) / T)^2 / r2
a1 = ((2 * π * r1) / T)^2 / r1
Given that r2 = 2 * r1, we can substitute this into the equations:
a2 = ((2 * π * (2 * r1)) / T)^2 / (2 * r1)
a1 = ((2 * π * r1) / T)^2 / r1
Simplifying these equations:
a2 = (4 * ((2 * π * r1) / T)^2) / (2 * r1)
a1 = ((2 * π * r1) / T)^2 / r1
The two r1 terms cancel out:
a2 = 4 * ((2 * π * r1) / T)^2 / (2 * r1)
a1 = ((2 * π * r1) / T)^2 / r1
Simplifying further:
a2 = (4 * π^2 * r1^2) / T^2
a1 = (4 * π^2 * r1^2) / T^2
We observe that the accelerations of the two satellites (a2 and a1) are equal. Therefore, the relationship between the accelerations of the satellites is that they are the same.