 # 1.) Kleppner &amp; Kolenkow, problem 7.6. [This is a tricky pro...

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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 𝑚, 𝑔, 𝑣, 𝑅. You can assume 𝑏 ≪ 𝑅. 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 𝐹⃑ = (𝑥2, 2𝑥𝑥). (a) Compute the work done by the force on a particle going from 𝑃 = (0, 0) to 𝑄 = (1,1) along the following 3 paths: i.) The straight segment along the x-axis from (0, 0) → (1,0), and the straight segment in the ydirection from (1,0) → (1, 1). ii.) Along the curve 𝑦 = 𝑥2. iii.) Along the curve given parametrically by (𝑥, 𝑦) = (𝑡3,𝑡2) 𝑓𝑓𝑓 0 ≤ 𝑡 ≤ 1. (b) Compute the work in going around the closed loop 𝑃 to 𝑄 along path (ii) above and then back to 𝑃 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) = 𝑉(𝑥) = 𝑈(𝑥) 𝑚 = 8𝑥 𝑥2+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 𝑥 < 0 approaches the origin with speed 𝑣0 = 4√2. 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 𝑥. 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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