A satellite is in a circular orbit around an unknown planet. The satellite has a speed of 1.53 x 104 m/s, and the radius of the orbit is 3.42 x 106 m. A second satellite also has a circular orbit around this same planet. The orbit of this second satellite has a radius of 9.28 x 106 m. What is the orbital speed of the second satellite?

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To determine the orbital speed of the second satellite, we can make use of the principle of conservation of angular momentum. According to this principle, the angular momentum of a satellite remains constant as long as no external torque acts on it.

Angular momentum (L) is defined as the product of moment of inertia (I) and angular velocity (ω).

L = I * ω

For a satellite moving in a circular orbit, the moment of inertia can be considered as the product of mass (m) and the square of the distance (r) from the axis of rotation.

I = m * r^2

Since the mass of the satellite remains constant, we can say that:

L1 = I1 * ω1
L2 = I2 * ω2

Since both satellites orbit the same planet, they experience the same gravitational force (F), and therefore, the same torque (τ) acting on them.

τ = r * F

As there is no external torque acting on the satellites, the torques cancel out. Hence:

τ1 = τ2

r1 * F1 = r2 * F2

The force acting on a satellite in circular motion can be given by the equation:

F = (m * v^2) / r

Therefore, we can rewrite the equation as:

r1 * (m * v1^2) / r1 = r2 * (m * v2^2) / r2

By canceling out the common terms, the equation simplifies to:

v1^2 = v2^2

Since the speed of the first satellite is given as 1.53 x 10^4 m/s, we can substitute this value into the equation to find the speed of the second satellite:

(1.53 x 10^4 m/s)^2 = v2^2

v2^2 = 2.3409 x 10^8 m^2/s^2

Taking the square root of both sides, we find:

v2 = 1.53 x 10^4 m/s

Therefore, the orbital speed of the second satellite is also 1.53 x 10^4 m/s.