Satellites: Jupiter completes one revolution about its own axis every 9.92 hours. What is the radius of the orbit required for a satellite to revolve about Jupiter with the same period? Jupiter has a mass of 1.90 × 10²⁷ kg and G = 6.67 × 10^-11 N ∙ m²/kg². A) 1.04 × 10⁷ m B) 2.26 × 10⁹ m C) 1.60 × 10⁸ m D) 3.41 × 10⁸ m E) 7.45 × 10⁸ m

Question

Quizplus · Accepted Answer

Using the formula for period of an orbit, we can find the required radius:
T = 2π(r/v)
Where T is the period (9.92 hours), r is the radius of the orbit we need to find, and v is the velocity of the satellite. The velocity can be found using the formula for centripetal force:
F = ma = mv²/r
v² = GM/r
v = √(GM/r)
Substituting this into the period formula gives:
T = 2π√(r³/GM)
Squaring both sides and rearranging gives:
r³ = (GMT²/4π²)
Substituting in the values for G, M, and T gives:
r³ = (6.67×10^-11 N∙m²/kg² × 1.90×10^27 kg × (9.92×3600 s)²)/(4π²)
Solving for r gives:
r = 1.60×10^18 m

Satellites: Jupiter Completes One Revolution About Its Own Axis Every