## Transcribed Text

1.) Kleppner & Kolenkow, problem 7.6. [This is a tricky problem... but you can do itl First, draw a clear
force diagram for the coin. You should be able to write down the magnitude of all of the forces in terms
of ๐, ๐, ๐ฃ, ๐
. You can assume ๐ โช ๐
. Also, recall that the total angular momentum is the sum of the
angular momentum of the center of mass plus the angular momentum around the center of mass. With
an appropriate choice of "pivot", you should be able to show that the angular momentum of the center
of mass around the pivot is constant, so now you only need to consider the angular momentum around
the center of mass. Be very careful with signs, especially with respect to which direction the angular
momentum vector is changing.]
2.) A mad scientist experimenting with quantum gravity has unfortunately created a rip in the fabric of
space-time, causing the Earth to suddenly have zero velocity. How long will it take for the Earth to hit
the surface of the Sun, and with what speed? (Ignore the motion of the Sun and assume it is fixed in
space.) When computing the time, you can ignore the radius of the Sun; but you cannot do that when
computing the velocity; explain why this is so.
3.) Consider the force ๐นโ = (๐ฅ2, 2๐ฅ๐ฅ).
(a) Compute the work done by the force on a particle going from ๐ = (0, 0) to ๐ = (1,1) along the
following 3 paths:
i.) The straight segment along the x-axis from (0, 0) โ (1,0), and the straight segment in the ydirection from (1,0) โ (1, 1).
ii.) Along the curve ๐ฆ = ๐ฅ2.
iii.) Along the curve given parametrically by (๐ฅ, ๐ฆ) = (๐ก3,๐ก2) ๐๐๐ 0 โค ๐ก โค 1.
(b) Compute the work in going around the closed loop ๐ to ๐ along path (ii) above and then back to ๐
along path (iii) in reverse. Do this in two ways:
i.) Using line integrals from part (a).
ii.) Using an area integral.
4.) Consider the 1-dimensional potential (potential energy per unit mass): U(x) fix V (x) =
๐(๐ฅ) = ๐(๐ฅ)
๐ = 8๐ฅ
๐ฅ2+1 .
(a) Graph this potential.
(b) Locate all points of equilibrium and classify their stability.
(c) Compute the period of small oscillations around any stable points of equilibrium.
(d) A particle initially located at very large at ๐ฅ < 0 approaches the origin with speed ๐ฃ0 = 4โ2. At what
location is the speed of the particle maximum? Compute the maximum speed of the particle.
(e) The particle inelastically collides with another particle of equal mass that is initially at rest at the
stable equilibrium point. What is the speed of the new combined particle immediately after the
collision? What fraction of the kinetic energy was converted to heat and sound during the collision?
(Hint: Energy is not conserved in inelastic collisions... but another quantity is conserved.)
(f) Is the new particle able to escape to positive infinity? If yes, what is its velocity for very large positive
๐ฅ. If it cannot escape to positive infinity, can it bounce back and escape to negative infinity? If yes, what
is its speed at negative infinity? If no, compute the range of at in which the particle oscillates.

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