ll Jazz LTE2:48 PM@ 76%( Classical-Dynamics-of-Particles-and-...PROBLEMS97245. Describe how to determine whether an equilibrium is stable or unstable when(d²U/dx²), = 0.246. Write the criteria for determining whether an equilibrium is stable or unstablewhen all derivatives up through order n, (d"U/dx"), = 0.247. Consider a particle moving in the region x>0 under the influence of the potentialU(x) = Upwhere U, = 1 J and a = 2 m. Plot the potential, find the equilibrium points, anddetermine whether they are maxima or minima.248. Two gravitationally bound stars with equal masses m, separated by a distance d, re-volve about their center of mass in circular orbits. Show that the period 7 is propor-tional to d/2 (Kepler's Third Law) and find the proportionality constant.2-49. Two gravitationally bound stars with unequal masses m, and mg, separated by a dis-tance d, revolve about their center of mass in circular orbits. Show that the period 7is proportional to d³/² (Kepler's Third Law) and find the proportionality constant.2-50. According to special relativity, a particle of rest mass mo accelerated in one dimen-sion by a force F obeys the equation of motion dp/dt = F Here p = mov/(1 –v/c*) !/2 is the relativistic momentum, which reduces to mov for v²/c² <1. (a) Forthe case of constant F and initial conditions x(0) = 0 = v(0), find x(t) and v(t).(b) Sketch your result for v(t). (c) Suppose that F/mo = 10 m/s² ( = g on Earth).How much time is required for the particle to reach half the speed of light and of99% the speed of light?2-51. Let us make the (unrealistic) assumption that a boat of mass m gliding with initialvelocity vo in water is slowed by a viscous retarding force of magnitude bu², where bis a constant. (a) Find and sketch v(t). How long does it take the boat to reach aspeed of w/1000? (b) Find x(t). How far does the boat travel in this time? Let m =200 kg, y = 2 m/s, and b = 0.2 Nm-2s2.2-52. A particle of mass m moving in one dimension has potential energy U(x) =U,[2(x/a) 2 - (x/a) *], where U, and a are positive constants. (a) Find the forceF(x), which acts on the particle. (b) Sketch U(x). Find the positions of stable andunstable equilibrium. (c) What is the angular frequency w of oscillations about thepoint of stable equilibrium? (d) What is the minimum speed the particle must haveat the origin to escape to infinity? (e) At t = 0 the particle is at the origin and its ve-locity is positive and equal in magnitude to the escape speed of part (d). Find x(4)and sketch the result.2-53. Which of the following forces are conservative? If conservative, find the potentialenergy U(r). (a) F, = ayz + bx + c, F, = axz + bz, F, = axy + by. (b) F, =- ze*, F, = Inz, F, = e=* + y/z. (c) F = e,a/r(a, b, c are constants).2-54. A potato of mass 0.5 kg moves under Earth's gravity with an air resistive force of- kmv. (a) Find the terminal velocity if the potato is released from rest and k =0.01 s-. (b) Find the maximum height of the potato if it has the same value of k,982/ NEWTONIAN MECHANICS–SINGLE PARTICLE

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Principles of Physics: A Calculus-Based Text

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Author:Raymond A. Serway, John W. Jewett

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Chapter28: Quantum Physics

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