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1. Gravity Problem (54 points total) NOTE: Do all calculations to 4 significant digits! Be sure to show all formulas and the numbers you are substituting. Include units in every answer. Constants: Mass of the Sun MO = 1.989x1030 kg G = 6.673x10-11 N-m2 /kg2 Comet Albatross (mass m = 8.600x1018 kg) orbits the Sun in an elliptical orbit (ε = 0.8200, semi-latus rectum α = 2.500x1011 m). Assume the Sun remains at rest. a) For this comet, find: (3 points each, 33 points) i. rmin (distance to the Sun at perihelion) ii. rmax (distance to the Sun at aphelion) iiii. a (semi-major axis of the orbit) iv. c (semi-focal distance) v. b (semi-minor axis of the orbit) vi. ETOT (total energy of the comet) vii. T (period of the orbit in years) viii. l ( angular momentum of the comet) ix. vp (speed at perihelion) x. va (speed at aphelion xi. dA/dt (rate at which the comet sweeps out area) b) You found rmin, the perihelion distance for comet Albatross. (3 points, 6 points total) i. If this value rmin is the radius of orbit for an asteroid orbiting the Sun, at what speed must the asteroid be moving? ii. If this value rmin is the perihelion distance for an object moving around the Sun in a parabolic orbit, at what speed must this object move? c) At point B, half-way between perihelion and aphelion (where the semi-minor axis meets the orbit), find: (5 points each, 10 points total) i.  ii. r c) At point C, the top of the latus rectum (just above the Sun), find: (3 points each, 6 points total) i.  ii. r B C Sun 2. Shrinking Pendulum (46 points total) A plane pendulum consists of a point mass m on a long string of length bo. At t = 0, the bob is released from rest at angle θo (measured from the string to the vertical y-axis). At the same time, the length of the string begins to decrease according to: b = boe -αt where α and bo are known constants. Assume: o The origin is at the point of support, the x-axis is horizontal, and U(0) = 0 o The pendulum support does not move, and there is no friction anywhere. o m remains in the x-y plane, and the string is always taut. o g = -gj. a) There is one degree of freedom: use θ (the angle between the string and the vertical) as the generalized coordinate. Start from expressions in Cartesian coordinates: (13 points) - Write an expression for the Lagrangian L = L(θ, ,t) for m - Write the Euler-Lagrange equation of motion. Simplify the equation, but do not solve. a) Again using θ as the generalized coordinate: (12 points) - Write the Hamiltonian H(θ,pθ,t) - Write the Hamilton equations of motion. Do not simplify or solve. b) Do the problem again, this time using Cartesian coordinates x and y as your two generalized coordinates. (13 points) - Write an expression for the Lagrangian L = L(x, x ,y, y ). - Now write the equation of constraint between x and y. - Write the two Euler-Lagrange equations of motion. Do not simplify or solve. c) For each: give an answer and a reason, not a calculation! (2 points each, 8 points) i. Is H = Etot? ii. Is pθ conserved? iii. Is ETOT conserved? iv. Is H conserved?

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