The planet Uranus has a radius of 25,560 km and a surface acceleration due to gravity of 11.1 m/s2 at its poles. Its m n Miranda (discovered by Kuiper in 1948) is in a circular orbit about Uranus at an altitude of 104,000 km above the planet’s surfaceMiranda has a mass of 6.6 × 1019 kg and a radius of 235 km. (a) Calculate the mass of Uranus from the given data. (b) Calculate the magnitude of Miranda’s acceleration due to its orbital motion about Uranus. (c) Calculate the acceleration due to Miranda’s gravity at the surface of Miranda. (d) Do the answers to parts (b) and (c) mean that an object released 1 m above Miranda’s surface on the side toward Uranus will tall up relative to Miranda? Explain.

Problem (a) To find the mass of Uranus M U Step 1: 7 Radius of Uranus R = U,560 km or 2.556 x 10 m 2 Acceleration due to gravity on the surface of uranus g = 11.Um/s Gravitational constant G = 6.67 x 10 -11Nm /kg 2 Altitude of Miranda above Uranus h = 104000 km or 1.04 x 10 m 8 5 Radius of Miranda R = M5 km or 2.35 x 10 m 19 Mass of Miranda M = 6M x 10 kg Step 2: We know that acceleration due to gravity on any planet having mass M and radius R GM g = R 2 Rearranging the above equation gR U 2 MU ----(1) G 7 2 11.1*(2.556 *0 ) MU 6.67 1011 * 25 26 MU 10.91 x 10 kg or 1.09 x 10 kg 26 The mass of Uranus M is 1.U x 10 kg Problem (b) To calculate the magnitude of Miranda’s acceleration on its orbit Step 1: This will be done by comparing Newton’s second law of motion to Newton’s gravitational force. GM U M F = M aM 2 ----(2) r Where M M is mass of Miranda and r is the distance between centers of Uranus and Miranda. Radius of miranda can be neglected...