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Chapter 13
Universal Gravitation (Homework)
Suggested Problems:
1. A 2 kg sphere and a 3 kg sphere have their centers 12 cm apart. What is the gravitational force
of one on the other?
Ans. 2.78 x 10 – 8 N
2. The moon’s mass is 7.35 x 10 22 kg and its radius is 1.74 x 10 6 m. If an astronaut is running on a
level surface and they launch themselves at 45 ο above the horizontal with a speed of 5 m/s. (a)
Calculate the acceleration due to gravity at the surface of the Moon? (b) How far from their
launch point do they land?
Ans. (a) 1.62 m/s 2 (b) 15.5 m
3. A satellite is orbiting 350 km above the surface of the Earth. (a) What is the acceleration due to
gravity the satellite experiences? (b) What is the period of the satellite?
Ans. (a) 8.83 m/s 2 (b) 5480 s
4. Two identical stars are in orbit about their center of mass. The orbital period is 141 days and
they are each moving at 29.6 km/s. Determine each star’s mass.
Ans. 6.04 x 10 30 kg
5. A synchronous satellite has an orbital period that is equal to the length of the planet’s day. If
we wish to place a synchronous satellite in orbit around Saturn what is the radius of its orbit?
(Saturn’s mass is 5.68 x 10 26 kg and it s day is 10.7 Earth hours long.)
Ans. 1.13 x 10 8 m
6. A 1000 kg object strikes the surface of the Earth. Assuming the object starts from rest an
infinite distance from the Earth how much energy is released in the collision?
Ans. 6.26 x 10 10 J
7. When the Sun dies it collapse down to a white dwarf about the size of the Earth but with 75 %
the mass of the Sun. (a) What is the density of the white dwarf? (b) What is the acceleration
due to gravity at its surface? (c) A 1000 kg object strikes the surface of the white dwarf.
Assuming the object starts from rest an infinite distance from the white dwarf how much
energy is released in the collision?
Ans. (a) 1.38 x 10 9 kg/m 3 (b) 2.45 x 10 6 m/s 2 (c) 1.56 x 10 16 J
8. 200 kg satellite is in orbit 400 km above the surface of the Earth. (a) How much energy must be
given to the satellite to move it to an orbit that is 600 km above Earth’s surface? (b) Calculate
the difference in the satellite’s kinetic energy. (c) Calculate the difference in the satellite’s
gravitational energy.
Ans. (a) 8.97 x 10 8 J (b) - 8.97 x 10 8 J (c) 1.79 x 10 8 J
Challenging Problems:
1. A spacecraft travels from the Earth to Mars using a minimum energy transfer orbit. This
means its perihelion position is the Earth’s orbit and its aphelion position is Mars’s orbit. How
much time does the trip take using a minimum energy transfer orbit?
Ans. 4.48 x 10 7 s or 518 days
2. A comet has a perihelion distance of 4 x 10 10 m and a period of 1500 years. What is the comet’s
aphelion distance?
Ans. 1.96 x 10 13 m or 131 AU
3. Taking into account the rotation of the Earth, what is the minimum launch speed required to
achieve orbit 400 km above the surface of the Earth if you launch from the equator?
Ans. 807 m/s