24. one point, from Lecture 24, short answers, tiny calculations, two things are "tricky"

(a) .5 Real-life Gamma Effects. All the gamma factor effects have been observed many, many times at particle accelerators where electrons, protons, or other tiny particles are accelerated to high gamma factors. One such particle is a mu meson (muon), which is just like an electron (negatively charged, etc.), except heavier. The muon has a normal lifetime of two-millionths of a second in its own rest frame.

A muon travels through a lab two kilometers long. The experimenter in the lab finds that the muon has a mass 2000 times its rest mass.
How long does the muon think the lab is? (.2)
How many seconds does the physicist say the muon lives in the lab? (.2) This is "tricky" because it turns the moving clock equation around so that you know the value on the moving clock. Just think about clocks running slower or faster.
How do you think the experimenter figured out the mass? (.1) This is "tricky" because I didn't say much if anything about it in lecture. It can be answered by putting some UNIT 1 physics together. Hint: it is "easy" to get the mass of a charged particle. Explain your answers and show your work.

(b) .5 What should it have been called? Einstein wanted to call it the Theory of Invariants instead of the Theory of Relativity. Name at least two invariant and three relative quantities in special relativity. (.1 apiece)

25. one point, from Lecture 25, a group question, short answers

(a) .2 Principle of Equivalence. State Einstein's Principle of Equivalence. (.1) Give an example which clearly illustrates it, explaining how it does so. (.1)

(b) .4 Principles of Inertia. State Newton's First Law. (.1) State Einstein's Principle of Inertia. (.2) Now relate the two to show that Newton's First Law is a special case of Einstein's Principle of Inertia, good only in flat spacetime. (.1)

(c) .4 Dynamics and Kinematics. In Unit 1 we first discussed classical kinematics then classical dynamics. Now we've gone through some relativity. Which of the two (kinematics or dynamics or both) is altered by special relativity and why? (.2) Which is or are altered by general relativity and why? (.2)

26. one point, from Lectures 25, 26, and 28, short answer, pictures, symbols

(a) .2 Geodesic. Give a general definition of a geodesic. (.1) Give the argument that light travels on a geodesic. (.1)

(b) .4 Mass Causes Curvature. Explain how mass causes the curvature of spacetime. This explanation must include how mass, dynamics, kinematics, and mathematics are all related. (.4) It is acceptable to use a diagram instead of a paragraph to answer this.

(c) .4 Gravitational Lensing. What is meant by gravitational lensing? (.1) How did a solar eclipse enable observation of this lensing? Draw the eclipse observation in two pictures, one before and one during the eclipse. (.2) (The second picture could be the one on p. 145 of the book.) What does gravitational lensing have to do with dark matter? (.1)

27. one point, from Lecture 27, a group question, short to medium answers

(a) .2 Einstein's Equation. What are the names and what is the meaning of T and R in Einstein's tensor equation -8(pi)[G/c² ]T=R?

(b) .8 Big Bang Cosmology. The four pillars of the theory of the Big Bang are the Hubble Expansion, the Cosmic Microwave Background Radiation (CMBR), Nucleosynthesis, and the structure of the CMBR. For each pillar, say what property of the universe it refers to and give a key observation which supports it. (.2 per pillar)

28. one point, from Lecture 28, short answer, some pictures, time consuming

(a) .3 Defining Open and Closed Universes. Define ``open'' and ``closed'' and ``just on the boundary'' universes in terms of expansion (draw this, relating it to a one-dimensional hyperbola, circle, and parabola), the curvature R, the mass density T, the mass density parameterOmega, and the total energy of the universe. There are 15 pieces of information here. (.1 for every 5 correct answers.)

(b) .3 Interesting Dead Stars. There are three end results for a ``dead'' star: a white dwarf, a neutron star, and a black hole. Say what each is in terms of initial mass relative to the mass of the sun, core mass relative to the mass of the sun, and final size (a length, not a mass). (.1 for every 3 correct answers.)

(c) .1 Defining a Black Hole. Characterize a black hole in terms of T, R, time, the path of light. (.1 if all four are right.)

(d) .2 Seeing Dark Things. How do we observe black holes (.1) and dark matter? (.1) Use a picture or two if you need to.

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