(1) Acceleration in Orbit
(3 pts.) A spacecraft en route to the International Space Station is orbiting the Earth in a circle. It is experiencing the Earth’s gravitational force and therefore by Newton’s Second Law it must be accelerating. The pilot notices that the spacecraft’s speed is constant. What is changing due to the Earth’s gravity?
(2) Retrograde Motion
(6 pts.) Suppose you hear that right now Mars is in retrograde motion, and that Jupiter is not in retrograde motion but instead is moving normally.
(a) Tonight, will Mars rise in the East or rise in the West?
(b) Tonight, will Jupiter rise in the East or rise in the West?
(3) Gravity on Another World
The planet Kenobi has a radius that is 0.77 times the Earth’s radius and a mass 0.48 times the Earth’s mass.
(3 pts.) (a) Calculate the ratio of the acceleration due to gravity on the surface of Kenobi compared with that on the surface of Earth. In other words, Kenobi’s gravity is what fraction of the strength of the Earth’s gravity? (Hint: Reviewing Unit 3.3 may be useful.)
(1 pt.) (b) Where would you weigh more–Earth or Kenobi?
(3 pts.) (c) A Kenobian called Ben weighs 322 Newtons on Kenobi–how much would Ben weigh on Earth?
(4) What is “weightlessness”?
The International Space Station generally operates in orbits at altitudes between 330-435 km above the Earth’s surface. You have probably seen astronauts float about “weightless” up there in “zero-g.”
(1 pts.) (a) Is the Earth’s gravity zero (or negligible) at the altitude of the space station’s orbit? Or is there significant gravity up there?
(2 pts.) (b) If the Earth’s gravity was zero at that altitude, what does Newton’s First Law tell you about how the shuttle would move?
(3 pts.) (c) Calculate the ratio of the acceleration due to the Earth’s gravity g on the surface of the Earth to g at 400 km above the Earth (RE = 6400 km).
(5) In orbit
Newton’s version of Kepler’s 3rd law (the most important equation in astrophysics!) allows us to measure the mass of anything, as long as it has something else orbiting it. It says that the mass of the object is proportional to the orbital size cubed and inversely proportional to the orbital period squared. Suppose you are on a starship exploring a new solar system, observing the orbits of moons around the new planets you’ve discovered…
(2 pts.) (a) Planet Kirk and planet Spock each have one moon, and both of these moons take exactly same amount of time to make one orbit. However Kirk’s moon orbits at a distance of 200,000 km and Spock’s moon orbits at a distance of 300,000 km. Which planet has more mass? Explain. (Note that I am asking you about the mass of planet Kirk and planet Spock. I am NOT asking you about the mass of Kirk’s moon or Spock’s moon.)
(3 pts.) (b) Planet Picard and planet Riker each have one moon, and both of these moons are the same distance away from their planets. However Picard’s moon takes less time to make an orbit. Which planet has more mass? Explain.
(3 pts.) (c) Suppose Picard’s moon takes three weeks to make one orbit, while Riker’s moon takes nine weeks to make one orbit. How many times more mass does your answer to part (b) have?
(3 pts.) (d) What does Newton’s version of Kepler’s 3rd law tell us about the mass of these moons? (The moons, NOT the planets.)
(6) Moons of Mars.
In 1727, Jonathan Swift wrote Gulliver’s Travels in which he described two moons orbiting Mars. About 150 years later, the American astronomer Asaph Hall used a large telescope to discover two moons of Mars and they were named Phobos (“fear”) and Diemos (“terror”). Their motions were surprisingly close to Jonathan Swift’s description. The orbital period of Phobos is 8 hours and that of Diemos about 30 hours.
(2 pts.) (a) Using the information given above and one of Kepler’s laws (which one?), explain which moon, Phobos or Diemos, is farther from Mars.
(2 pts.) (b) Describe the motions of Phobos and Diemos as seen by an inhabitant of Mars at, say, 40Â° north martian latitude.
(Note: Phobos and Diemos both orbit Mars close to Mars’ equatorial plane and in the same (counterclockwise) direction that Mars spins. Mars’ spin period is close to 24 hours. Drawing diagrams would probably help).
(7) Mass and Gravity
(3 pts.) In the morning, a man steps on a bathroom scale and finds that he weighs 150 pounds. At noon we magically double the mass of every object in the universe. If he steps on the scale before bed, how much will he weigh?