Trying to find orbital radius of a satellite

[masterofphys]

Homework Statement

A geostationary satellite in orbit around the Earth has a period identical to that of the Earth's daily rotation; the radius of such an orbit is 4.23 \times 10^4 kilometers. A system of satellites is proposed such that 15 satellites are in orbit at the same radius and each passes over a point fixed on the Earth 14 times per day (the times will be evenly spaced). What will be the orbital radius of each satellite? There will be two such radii; only the smaller will be practical.

Homework Equations

V = $$\sqrt{}Gm(earth)/r$$
T = 2pir/v

The Attempt at a Solution

I tried finding what the current speed would be, but I got a larger value. I'm assuming the speed has to be 14 times what it originally is. I don't know where to begin!

 

[Redbelly98]

Rather than thinking what the speed should be, try to figure out what either the angular speed or the orbital period should be.

 

[thomate1]

As a scientist, it is important to have a clear understanding of the concepts and equations involved in solving a problem. In this case, the first step would be to determine the period of the proposed satellites. Since each satellite passes over a fixed point on Earth 14 times per day, the period would be 1/14 of a day, or approximately 1.15 hours.

Next, we can use the equation for orbital period, T = 2πr/v, to calculate the orbital velocity of the satellites. We know the period (T) and the radius (r) from the given information, so we can rearrange the equation to solve for v.

v = 2πr/T = 2π(4.23 x 10^4 km)/(1.15 hours) = 2.93 x 10^4 km/h

This is the velocity at which the satellites would need to orbit in order to pass over a fixed point on Earth 14 times per day. However, this is not the final answer as we need to find the orbital radius of each satellite.

To do this, we can use the equation for orbital velocity, V = √(Gm/r), where G is the gravitational constant, m is the mass of the Earth, and r is the orbital radius. We know the velocity (V) from the previous calculation, and we can assume that the mass of the Earth (m) remains constant. Therefore, we can rearrange the equation to solve for r.

r = Gm/V^2 = (6.67 x 10^-11 Nm^2/kg^2)(5.97 x 10^24 kg)/(2.93 x 10^4 km/h)^2 = 4.27 x 10^4 km

This is the orbital radius of each satellite in the proposed system. However, the question states that there will be two such radii, and only the smaller one will be practical. To find the smaller radius, we can use the fact that there are 15 satellites in orbit at the same radius. Therefore, the total length of the orbit (2πr) must be evenly divisible by 15.

2πr = n(15) = 15n, where n is any positive integer.

To find the smallest value of r that satisfies this condition, we can set n = 1, and solve for r.

r = (15n)/(