intro_physics_1

Week 12: Gravity 623 Problem 5. r ω MR This problem will help you learn required concepts such as: • Newton’s Law of Gravitation • Circular Orbits • Centripetal Acceleration • Kepler’s Laws so please review them before you begin. A straight, smooth (frictionless) transit tunnel is dug through a spherical asteroid of radius R and mass M that has been converted into Darth Vader’s death star. The tunnel is in the equatorial plane and passes through the center of the death star. The death star moves about in a hard vacuum, of course, and the tunnel is open so there are no drag forces acting on masses moving through it. a) Find the force acting on a car of mass m a distance r < R from the center of the death star. b) You are commanded to find the precise rotational frequency of the death star ω such that objects in the tunnel will orbit at that frequency and hence will appear to remain at rest relative to the tunnel at any point along it. That way Darth can Use the Dark Side to move himself along it almost without straining his midichlorians. In the meantime, he is reaching his crooked fingers towards you and you feel a choking sensation, so better start to work. c) Which of Kepler’s laws does your orbit satisfy, and why?

624 Week 12: Gravity Problem 6. mN r0= R ρ 0 S This problem will help you learn required concepts such as: • Newton’s Second Law. • Newton’s Law of Gravitation • Gravitational Field/Force Inside a Spherical Shell or Solid Sphere. • Harmonic Oscillation Given Linear Restoring Forces. • Definitions and Relations Involving ω and T . so please review them before you begin. A straight, smooth (frictionless) transit tunnel is dug through a planet of radius R whose mass density ρ0 is constant. The tunnel passes through the center of the planet and is lined up with its axis of rotation (so that the planet’s rotation is irrelevant to this problem). All the air is evacuated from the tunnel to eliminate drag forces. a) Find the force acting on a car of mass m a distance r < R from the center of the planet. b) Write Newton’s second law for the car, and extract the differential equation of motion. From this find r(t) for the car, assuming that it starts at rest at r0 = R on the North Pole at time t = 0. c) How long does it take the car to get to the South Pole starting from rest at the North Pole? How long does it take to get back to the North Pole? Compare this (second answer) to the period of a circular orbit inside the death star you found (disguised as ω) in the previous problem. d) A final thought question: Suppose it is released at rest from an initial position r0 = R/2 (halfway to the center) instead of from r0. How long does it take for the mass to get back to this point now (compare the periods)?

Week 12: Gravity 625 All answers should be given in terms of G, ρ0, R and m.

626 Week 12: Gravity Problem 7. Ueff E 3 r E 2 E1 E0 The effective radial potential of a planetary object of mass m in an orbit around a star of mass M is: L2 GM m 2mr2 r Ueff (r) = − (a form you already explored in a previous homework problem). The total energy of four orbits are drawn as dashed lines on the figure above for some given value of L. Name the kind of orbit (circular, elliptical, parabolic, hyperbolic) each energy represents and mark its turning point(s).

Week 12: Gravity 627 Problem 8. In a few lines prove Kepler’s third law for circular orbits around a planet or star of mass M : r3 = CT 2 and determine the constant C and then answer the following questions: a) Jupiter has a mean radius of orbit around the sun equal to 5.2 times the radius of Earth’s orbit. How long does it take Jupiter to go around the sun (what is its orbital period or “year” TJ )? b) Given the distance to the Moon of 3.84 × 108 meters and its (sidereal) orbital period of 27.3 days, find the mass of the Earth Me. c) Using the mass you just evaluated and your knowledge of g on the surface, estimate the radius of the Earth Re. Check your answers using google/wikipedia. Think for just one short moment how much of the physics you have learned this semester is verified by the correspondance. Remember, I don’t want you to believe anything I am teaching you because of my authority as a teacher but because it works.

628 Week 12: Gravity Problem 9. It is very costly (in energy) to lift a payload from the surface of the earth into a circular orbit, but once you are there, it only costs you that same amount of energy again to get from that circular orbit to anywhere you like – if you are willing to wait a long time to get there. Science Fiction author Robert A. Heinlein succinctly stated this as: “By the time you are in orbit, you’re halfway to anywhere.” Prove this by comparing the total energy of a mass: a) On the ground. Neglect its kinetic energy due to the rotation of the Earth. b) In a (very low) circular orbit with at radius R ≈ RE – assume that it is still more or less the same distance from the center of the Earth as it was when it was on the ground. c) The orbit with minimal escape energy (that will arrive, at rest, “at infinity” after an infinite amount of time). Problem 10. d y mx M = 80m D = 5d The large mass above is the Earth, the smaller mass the Moon. Find the vector grav- itational field acting on the spaceship on its way from Earth to Mars (swinging past the Moon at the instant drawn) in the picture above.

Week 12: Gravity 629 Problem 11. Ma Re Me This problem will help you learn required concepts such as: • Gravitational Energy • Fully Inelastic Collisions so please review them before you begin. A bitter day comes: a roughly spherical asteroid of radius Ra and density ρ is discov- ered that is falling in from far away so that it will strike the Earth. Ignore the gravity of the Sun in this problem. Determine: a) If it strikes the Earth (an inelastic collision if there ever was one) how much energy will be liberated as heat? Express your answer in terms of Re and either g or G and Me as you prefer. It is probably very safe to say that Ma ≪ Me... b) Chuck Norris lands on the surface of the asteroid to save the Earth, but instead of screwing around with drills and nuclear bombs Chuck jumps up from the surface of the asteroid at a speed of vcn to deliver a roundhouse kick that would surely break the asteroid in half and cause it to miss the earth – if it knows what’s good for it (this is Chuck Norris, after all). However, if you jump up too fast on an asteroid, you don’t come down again! Does Chuck ever fall down onto the asteroid after his jump? c) Evaluate your answers to a-b above for the following data: ρ = 6 × 103 kilograms/meter3 (12.58) Ra = 104 meters Re = 6.4 × 106 meters g = 10 meters/second2 Me = 6 × 1024 kilograms vcn = 5 meters/second Express your answer to a) both in joules and in “tons” (of TNT) where 1 ton-of-TNT = 4.2 × 109 joules. Compare the answer to (say) 30 Gigatons as a safe upper bound for the total combined explosive power of every weapon (including all the nuclear weapons) on earth.

630 Week 12: Gravity d) Find the size of an asteroid that (when it hits) liberates only the energy of a typical thermonuclear bomb, 1 megaton of TNT. Problem 12. There is an old physics joke involving cows, and you will need to use its punchline to solve this problem. A cow is standing in the middle of an open, flat field. A plumb bob with a mass of 1 kg is suspended via an unstretchable string 10 meters long so that it is hanging down roughly 2 meters away from the center of mass of the cow. Making any reasonable assumptions you like or need to, estimate the angle of deflection of the plumb bob from vertical due to the gravitational field of the cow.

Week 12: Gravity 631 Optional Problems The following problems are not required or to be handed in, but are provided to give you some extra things to work on or test yourself with after mastering the required problems and concepts above and to prepare for quizzes and exams. Continue Studying for Finals using problems from the online review!