Figure 149: A typical energy diagram illustrating the e ective potential energy, which is basically the sum of the radial potential energy and the angular kinetic energy of the orbiting object.

By drawing a constant total energy on this plot, the di erence between Etot and Ue (r) is the radial kinetic energy, which must be positive. We can determine lots of interesting things from this diagram.

~

In figure 150, we show orbits with a given fixed angular momentum L 6= 0 and four generic total energies Etot. These orbits have the following characteristics and names:

a)Etot > 0. This is a hyperbolic orbit.

b)Etot = 0. This is a parabolic orbit. This orbit defines escape velocity as we shall see later.

c)Etot < 0. This is generally an elliptical orbit (consistent with Kepler’s First Law).

d)Etot = Ue ,min. This is a circular orbit. This is a special case of an elliptical orbit, but deserves special mention.

Note well that all of the orbits are conic sections. This interesting geometric connection between 1/r2 forces and conic section orbits was a tremendous motivation for important mathematical work two or three hundred years ago.

12.7: Escape Velocity, Escape Energy

As we noted in the previous section, a particle has “escape energy” if and only if its total energy is greater than or equal to zero, provided that we set the zero of potential energy at infinity in the first place. We define the escape velocity (a misnomer!) of the particle as the minimum speed (!) that it must have to escape from its current gravitational field – typically that of a moon, or planet, or star. Thus:

Just to whet your interest (and explain why we have spent so long on gravity when it is weak and mostly irrelevant outside of its near-Earth form in everyday a airs) let is take note of Coulomb’s Law, the force that governs the all-important electrostatic interaction that binds electrons to atomic nuclei to make atoms, and binds atoms together to make molecules. It is the force that exists between two charges, and can be written as:

Hmmm, this equation looks rather familiar! It is almost identical to Newton’s Law of Gravitation, only it seems to involve the charge (q) of the particles involved, not their mass, and an electrostatic constant ke instead of the gravitational constant G.

In fact, it is so similar that you instantly “know” lots of things about electrostatics from this one equation, plus your knowledge of gravitation. You will, for example, learn about the electrostatic field, the electrostatic potential energy and potential, you will analyze circular orbits, you will analyze trajectories of charged particles in uniform fields – all pretty much the same idea (and algebra, and calculus) as their gravitational counterparts.

The one really interesting thing you will learn in the first couple of weeks is how to properly describe the geometry of 1/r2 force laws and their underlying fields – a result called Gauss’s Law. This law and the other Maxwell Equations will turn out to govern nearly everything you experience. In some very fundamental sense, you are electromagnetism.

Good luck!

Homework for Week 12

Problem 1.

Physics Concepts: Make this week’s physics concepts summary as you work all of the problems in this week’s assignment. Be sure to cross-reference each concept in the summary to the problem(s) they were key to, and include concepts from previous weeks as necessary. Do the work carefully enough that you can (after it has been handed in and graded) punch it and add it to a three ring binder for review and study come finals!

Problem 2.

It is a horrible misconception that astronauts in orbit around the Earth are weightless, where weight (recall) is a measure of the actual gravitational force exerted on an object. Suppose you are in a space shuttle orbiting the Earth at a distance of two times the Earth’s radius (Re = 6.4 × 106 meters) from its center.

a)What is your weight relative to your weight on the Earth’s surface?

b)Does your weight depend on whether or not you are moving at a constant speed? Does it depend on whether or not you are accelerating?

c)Why would you feel weightless inside an orbiting shuttle?

d)Can you feel as “weightless” as an astronaut on the space shuttle (however briefly) in your own dorm room? How?

Problem 3.

Physicists are working to understand “dark matter”, a phenomenological hypothesis invented to explain the fact that things such as the orbital periods around the centers of galaxies cannot be explained on the basis of estimates of Newton’s Law of Gravitation using the total visible matter in the galaxy (which works well for the mass we can see in planetary or stellar context). By adding mass we cannot see until the orbital rates are explained, Newton’s Law of Gravitation is preserved (and so are its general relativistic equivalents).

However, there are alternative hypotheses, one of which is that Newton’s Law of Gravitation is wrong, deviating from a 1/r2 force law at very large distances (but remaining a central force). The orbits produced by such a 1/rn force law (with n =6 2) would not be elliptical any more, and r3 =6 CT 2 – but would they still sweep out equal areas in equal times? Explain.

Problem 4.

r

ω

M

R

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?

Problem 6.

The e ective radial potential of a planetary object of mass m in an orbit around a star of mass M is:

(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).

512 Week 12: Gravity

Problem 7.

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.

Problem 10.

M a

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 discovered 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:

g= 10 meters/second2

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.

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 11.

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.

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!