Physics 264L: Some Review Questions for the First Midterm
Note: Don’t try to do all of these problems. Instead, do a few problems to help strengthen where you might
have a weaker understanding, and or do a few problems to help you review key arguments or equations. But
best is just to think about how to approach the problems, rather than solve them in their entirety.
1. Extra problems from the French text:
(a) 1-7 (good order-of-magnitude estimatation problem)
(b) 1-15 (practice with the relativistic kinetic energy formula)
(c) 1-19 (you did this, but review for practice with binomial approximations).
(d) 1-24
(e) 4-2
(f) 4-4 (another good order-of-magnitude estimation problem)
(g) 4-6
(h) 4-7
(i) 4-8
(j) 4-9
(k) 4-16
(l) 5-1
(m) 5-2
(n) 5-4
(o) 5-12
(p) 5-17
(q) 6-2 (practice with the Lorentz transformation for E/c2 and p, and the fact that E 2 − (pc)2 =
(m0 c2 )2 in all inertial frames of reference.
(r) 6-3
(s) 6-10
(t) 7-7
2. From chapters 1 and 2 of the posted chapters of Tipler and Llewellyn, 5th edition:
(a) 1-6
(b) 1-8
(c) 1-18
(d) 1-23
(e) 1-34
(f) 1-47
(g) 1-58
(h) 2-12
(i) 2-42
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3. Some miscellaneous questions, to stimulate your thinking:
(a) If two triangles at rest are congruent in inertial frame S, will they still be congruent in all other
inertial frames S’ ?
(b) If four points such that no three points are collinear are at rest and lie on a plane in an inertial
frame S, will they lie on a plane for all other inertial frames S’, i.e., is “lying on a plane” a
relativistic invariant property?
(c) If E is the relativistic mass-energy and p are the relativistic mass-energy and momentum of a
particle in a certain frame S, use the Lorentz transformation for energy and momentum to show
that the quantity E 2 − (pc)2 is a relativistic invariant that has the same value in all inertial frames
of reference.
(d) Using the formula
2πr − C
3
lim
,
(1)
π r→0
r3
for the curvature K of space at some point in terms of geodesic radial lengths r and the measured
circumference C of a circle of radius r, calculate the non-zero curvature at the center of a merrygo-round
that is rotating with constant angular speed ω and estimate the radius of curvature
√
1/ K for a typical maximum speed of rotation of such a merry-go-round.
K=
(e) Assuming a proton and electron are point classical particles with the electron orbiting the proton
in a circular orbit, estimate to the nearest power of ten the orbital radius such that the electrical
binding energy of the hydrogen atom has an inertial mass equal to the mass of the proton.
(f) Given two photons of frequencies f1 and f2 , in what direction should you direct one photon
relative to the other so that the 2-photon system has the largest possible rest mass?
(g) A pendulum of length L with mass M hangs from the ceiling of a car. If the car now accelerates
with acceleration a, use the equivalence principle to deduce what angle the pendulum will make
with the vertical in steady state.
(h) Two sticks of length L are parallel to the x axis and bisected by the y axis of an inertial frame
of reference, and these sticks travel in the +y direction with speeds v1 and v2 < v1 (so the slower
stick is behind and never catches up to the faster stick). True or false and why: the two stickes
will remain parallel in any other inertial frame of reference.
(i) True or false and why: if a rocket moves with constant speed along the line y = vt+x0 with x0 ̸= 0
(so the rocket does not pass through the origin of the inertial frame S), then the amount of time
dilation that occurs for time intervals measured in the rocket will depend on the location of the
rocket.
(j) True or false and why: if a thin uniform rod has a moment of inertia I0 about its center when at
room temperature, and if I is its moment of inertia on a hot summer day, then I > I0 .
(k) Does a photon have kinetic energy? If so, what is its value?
(l) Explain why, according to special relativity, a particle with zero rest mass must move at the speed
of light c.
(m) Explain how to use stellar aberration to deduce a value for the speed of light from observations
of stars and knowledge of Earth’s orbit around the Sun.
(n) Consider a Michelson-Morley experiment with arms parallel to the x and y axes of an inertial
frame of reference S and that moves with constant speed v along the +x axis. Calculate the
difference in time ∆t1 − ∆t2 for light to traverse both arms as viewed by a stationary observer
in S.
(o) Give two examples of non-inertial frames of reference.
(p) When a metal rod undergoes relativistic contraction as it moves by a given observer, is there a
physical increase in the stress of the rod, as if forces were applied to the ends of the rod?
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