Physics 10: Final Exam Study Guide

Spring Term, 2008

The final exam will cover lecture material from May 7 (Lecture 13 on Rockets and Gravity) through the last lecture (June 6, Lecture 24 on the Frontiers of Physics). While the exam is not cumulative in the strict sense, you will need to remember some of the basic concepts from the first half, like force, work, energy, power. These concepts have continued to play a role in the second half of the course.

You may also want to study the transmitter questions we saw in class. These will be posted on the Lectures page (look for the last entry). Note also the first-half questions after Lecture 11 on the same page.

Below are the topics that are likely to appear in some form on the exam. The exam will consist of multiple choice, true/false, and short answer. I'll give you a bucket of equations on the front page of the exam, though fewer will be needed than on the first exam, due to the more qualitative nature of the second half of the course.

• The speed of light = 300,000,000 m/s = 3×10⁸ m/s = 300,000 km/s (always remember this one—who knows when it'll crop up)
• Review the major concepts from the first half of the course—concepts that have stayed with us in the second half, such as force, acceleration, energy, power, etc. The questions on the exam will not focus on these concepts in the way that the first exam did, but the fundamental concepts will be used.
• A rocket pushes against its own fuel. It requires no other medium (air, water, land) to push against. In effect, the rocket pushes against the inertia of the fuel—in such a way as to conserve momentum. With a given mass available in fuel, you do better flinging many small particles fast than a few big particles slow. On a frictionless sled on an icy lake, 25 pounds of BBs (and associated gun) will serve better than a single 25 pound brick to get you going. In real life, friction would render the BBs useless, but in space this would certainly be your best bet.
• A circular orbit results when the centripetal acceleration, a = v²/r, is provided by gravity. Set a = g = 9.8 m/s² to get the condition for low-earth orbit. In effect, the curvature of the earth is such that the earth falls away beneath the orbiting satellite as it continuously "falls around" the earth.
• The sensation of weightlessness in orbit is not because you are far from earth's gravity—this is far from true. Gravity is responsible for bending the path into an orbit! The weightlessness is because both the astronaut and the craft are falling toward the earth at the same acceleration, so that there is no relative acceleration between astronaut and craft.
• Newton's law of universal gravitation states that the force on a mass, m, due to mass M a distance r away is: F = GMm/r², where G is the gravitational constant (6.67×10^-11 in MKS—you don't need to remember this number). Understand qualitatively how the force varies if we change masses and/or distances.
• Since F = ma, the acceleration of mass, m due to M is a = GM/r², and is independent of the mass (why a bowling ball and golf ball accelerate the same).
• You can figure any circular orbit by setting centripetal acceleration equal to gravitational acceleration: v²/r = GM/r², reducing to v² = GM/r.
• Geosynchronous orbit results when the orbital period comes out to equal 24 hours. The satellite is still in orbit, though from our position on the rotating earth the satellite appears to hover motionless in the sky. Think of twirling with a ball on a string in front of you. You see the ball always in front, but outsiders know that the ball is really making circles around you.
• The requirement that the speed of light is measured to be the same for all observers regardless of their state of motion produces a variety of consequences, collectively referred to as special relativity.
• There is no well-defined notion of simultaneity in special relativity: observers moving relative to each other will not agree on whether two events are simultaneous. Space and time get mixed together into what we call spacetime.
• Time appears to slow (but never reverse) for objects moving relative to the observer. The amount of slowing goes by the ratio γ (gamma).
• Length appears to contract along the direction of travel for objects moving relative to the observer. The "shrinkage" also goes by the ratio γ.
• At familiar speeds, γ is 1.0. At 0.6c, γ = 1.25. At 0.87c, γ = 2.0. As the velocity approaches the speed of light, γ approaches infinity. No need to remember specific numbers—just the endpoints and the general trend.
• Example: at 0.87c, γ = 2.0, so you see a meter stick with a watch attached moving past you at this speed appear to be only 0.5 meters long (stick is oriented along direction of travel), and the second hand only ticks once every two seconds (by your own reckoning of time). Somewhat non-intuitively, an observer traveling with that meter stick and clock sees your meter stick to be only 0.5 meters long in the direction of travel, and your clock appears to tick slowly according to their clock. The relativity of motion (think empty space) makes it impossible to decide who is at rest and who is in a state of motion—it's relative motion that's important.
• Be able to use the velocity addition rule to add two velocities in a manner consistent with special relativity. This rule is V = (v₁ + v₂)/(1 + v₁v₂/c²). Note the ninth edition of the Hewitt book has a sign error in the denominator on p. 703.
• Einstein wanted a reason for gravitation, and relegated it to the status of a "fictitious" force: an artifact of not being in a freely falling frame. In effect, while standing on the earth's surface, the natural frame of reference falls past us at 9.8 m/s². Life in a falling elevator is life in a naturally falling frame of reference. This immediately explains why different objects experience the same acceleration in a gravitational field. Think in terms of the accelerating box in space: it's the box that's accelerating, not the objects "falling" within the box. But from within the box, it looks like all objects accelerate toward the floor at the same rate. They're just trying to move with uniform velocity, in accordance with Newton's first law. The floor comes up to hit them.
• Out of General Relativity came the notion of curved spacetime. This naturally explained mass attraction as a response to curved space. Just like bowling balls on a mattress attract each other, dimples of curved spacetime also find each other attractive.
• The prescription of general relativity is: the presence of mass curves spacetime, and spacetime tells mass how to move. Specifically, falling/orbiting bodies simply follow the straightest lines they possibly can through the curved spacetime.
• General Relativity makes definite predictions that allow it to be rigorously tested. The three classic tests are the perihelion precession of Mercury's orbit, the deflection of starlight (stars near sun appear to be pushed farther from sun's edge), and the gravitational redshift (time runs slower in presence of gravity). So far General Relativity has passed all tests with flying colors, though its fundamental incompatibility with quantum mechanics motivates further tests.
• Electrons are negatively charged, and protons are positive by the exact same amount. Like charges repel, and unlike charges attract.
• Electric charges are transferred by rubbing, nearly always due to one surface grabbing electrons off the other surface. Which material donates electrons and which grabs them is not usually evident (and almost impossible to guess). Electrons are the transferred charges because electrons occupy the vulnerable outskirts of atoms, and are therefore easily grabbed.
• Review the physics of sparks and lightning; the breakdown threshold (3 million volts per meter), and how this scales to smaller lengths.
• Understand the electrostatic force law and how the force scales with the various parameters (e.g., double one charge, half the separation, etc.). Appreciate the similarity between this law and Newton's law of gravity.
• The electric field is a concept that not only tells you the direction of force a charge will experience, but also the magnitude of the force. Specifically, F = qE, where q is the charge (in Coulombs), F is the force vector (has magnitude, in Newtons, and direction), and E is the electric field vector (has magnitude, in Volts per meter, and direction).
• Electric field lines point from positive to negative charges. An electron, being of negative charge, is attracted to positive charge, and thus feels force against the direction of the field lines. A positive charge feels force along the direction indicated by the ambient field lines.
• Because the electric field mirrors the force, the electric field from a charge diminishes as the square of the distance, just like the force does.
• Electric current is simply the motion of charges along a particular direction.
• Review the relationship between electricity and magnetism. Specifically, in what way can electric charges produce magnetic fields, and in what way can magnets produce electric fields (that then move charges)?
• Be able to use the relation (frequency in Hz)*(wavelength in meters) = (speed of light in m/s). Know also that radio antennas are optimally one-quarter wavelength long.
• Know about the properties of electromagnetic radiation: generated by accelerating charges (electrons); consists of oscillating electric and magnetic fields at 90-degrees to each other; a source can be polarized so that every wave has the same orientation (e.g., vertical) of the electric field vector; encompasses radio, microwave, infrared, visible, ultraviolet, X-ray, and gamma radiation. All forms of electromagnetic radiation travel at precisely the speed of light (in a vacuum).
• Appreciate the pre-quantum problems in physics that motivated quantum mechanics. Specifically, understand the argument about atoms decaying in mere nanoseconds and also the mystery of atomic spectra.
• The energy of a single photon of light is E = hν, or E = hf, where ν (nu) and f are two equivalent ways of labeling the frequency. h is Planck's constant, and is 6.626×10^-34 J·s. Ultraviolet photons are more damaging because each photon packs a greater punch—enough to bust up molecules in some cases.
• Light is both a particle and a wave. But it can behave like both, or more generally like one at a time. The bottom line is that there's a wave-particle duality—light comes with both properties simultaneously. Moreover, all particles have a wave-like nature, characterized by the de Broglie wavelength: λ = h/p, where h is Planck's constant, and p = mv is the particle's momentum.
• The Heisenberg Uncertainty Principle states, in one form, that it is impossible to simultaneously achieve arbitrary precision in knowing both the position and momentum (velocity) of a particle. It's not a matter of poor measurement techniques, but stems from the fact that the act of localizing the particle precisely ends up imparting some unknown momentum to the particle in question—effectively adding uncertainty to the particle's momentum.
• Diffraction is an example of the uncertainty principle in action. Light that goes through a tiny opening is localized very well, so that its momentum in the direction/dimension of localization is poorly known. Thus it spreads. The tighter the aperture, the greater the spread in angle.
• The quantum view of electrons in atoms (e.g., hydrogen) is not one of electrons whizzing about in orbits, but rather of static (stationary) probability distributions describing where one might find the electron if one tried to localize it. A variety of possible configurations exist, each having a distinct energy value. The steps in energy explain the atomic spectra, and why each element's spectrum is different (different sets of levels in each element). The stationary aspect of the electron distribution solves the problem of atom decay: the electrons are not accelerating around the nucleus in orbits, which would require that they lose energy by electromagnetic radiation and spiral into the nucleus.
• Know the wavelength range associated with visible light.
• Understand why the sky is blue and by association, why sunsets are red/orange.
• Understand color combination: if R, G, B represent red, green, and blue, C, M, Y represent cyan, magenta, yellow, and W means white, then:
  • R + G + B = W;
  • R + G = W − B = Y;
  • R + B = W − G = M;
  • G + B = W − R = C.
• The additive laws cover color addition, as one gets by adding light sources (TV, computer screen, LEDs, light bulbs). The subtractive laws deal with absorption by paints and dyes. So mixing yellow and magenta paints is basically combining blue and green absorbers, so red should result.
• Understand the geometry of rainbows: specifically where in relation to the sun one must look to find a rainbow.
• Despite their mutual electric repulsion, the positive charges in a nucleus are bound together by a more powerful attractive force: the (short-range) strong nuclear force.
• A given isotope of an element (X) has in its nucleus A nucleons consisting of Z protons and N neutrons, such that A represents the total number: A = Z + N. We denote this element as ^AX. For example, uranium-238 (²³⁸U) has 92 protons and 146 neutrons for a total of 238 nucleons.
• Uranium and plutonium are special, in that these are the only two elements with isotopes that can undergo fission via the introduction of an extra wandering neutron.
• Before and after nuclear reactions, the total number of nucleons hasn't changed, but their rearrangement releases energy pent up in the binding energy of the nucleus. The end product is always less massive than the initial fuel, and this mass difference has produced E = mc² of energy. For fission, the products are about 0.1% less massive than the starting mass.
• Iron is at the peak of the nuclear binding energy curve. This means on the light side of iron, fusion pays off, and on the heavier side, it's fission. We'll never build plants for fission of light elements: this would result in a loss of energy. Likewise we (and also stars during normal operation) will not fuse beyond iron.
• Fusion would be a fantastic source of energy: nearly unlimited supply of raw material, almost no radioactive by-products, and far more efficient than fission, even. But it's challenging to make a 50 million degree contained plasma. And it always seems to be 50 years in the future.
• Basic research often has unanticipated payoffs. Shockley wasn't trying to build a computer when he invented the transistor. The world-wide-web sprung out of scientists trying to distribute information over DARPA-funded communications lines.
• There are four fundamental forces in nature that we know about: gravity, electromagnetism, and the strong and weak nuclear forces. We have united the latter three under the banner of quantum mechanics, but gravity remains the odd-man-out. It is still unclear what further unifications may be forthcoming.
• Many of our advances in physics have come from new appreciations about the nature of space and time. This trend is likely to continue into the future.
• Science is driven by questions and exploration. Never content to blindly accept the current theories, scientists are always challenging assumptions—pushing at the edges and exposing the weaknesses of the current state of affairs. The final arbiter of scientific dispute is experiment. Science is forced to "follow the data"—even if it leads in new and strange (and unwanted) directions.