Note that many Barger & Olsson problems have final answers in the back of the book, so you can check your work!

Reading:

Read B & O Chapter 1 -- 1.6; Symon 2.2 -- 2.6; KB 361 lecture notes 1 -- 3.

Problems for Review:

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1. 1. A particle of mass m, initially at rest, moves one-dimensionally subject to a net force . Integrate Newton's second law to find its position and velocity as functions of time, and plot them (qualitatively).
   2. For part (a) above, express the force as a function of velocity. Assume this force describes the motion of a ball dropped, from rest, into a viscous fluid. What are appropriate values for the constants a and b, in terms of the gravitational acceleration g and the ball's terminal velocity ?

2. A mass m sliding horizontally is subject to a viscous drag force. The mass has initial velocity (at x=t=0) and a retarding force . Using the methods of sections 1.4 and 1.6 for velocity dependent force, find
   1. the velocity and position as functions of time -- v(t), x(t). Use this result to solve for velocity as a function of distance, v(x).
   2. Rearrange and integrate the equation of motion directly to find v(x). (This should agree with (a).)
   3. Show that the mass never comes to rest.
3. The speed of a particle of mass m varies with distance as . Assume the particle starts at rest at the origin at t=0.
   1. Find the force F(x) responsible.
   2. Determine x(t) and
   3. F(t).

Problems to Solve:

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1. As in review problem 2, a mass m slides horizontally subject to a viscous drag force. Here, for an initial velocity (at x=t=0) and a retarding force , find
   1. the velocity and position as functions of time -- v(t), x(t). Use this result to solve for velocity as a function of distance, v(x).
   2. Rearrange and integrate the equation of motion directly to find v(x). (This should agree with (a).)
   3. Show that the mass moves a finite distance before coming to rest.
2. B & O 1-9.
3. A perfectly flexible cable has length l. Initially, the cable is at rest, with a length of it hanging vertically over the edge of a table. Neglecting friction, consider the cable's motion as it slips off the edge of the table.
   1. Derive an equation of motion for the length of cable x hanging over the edge of the table after a time t. Assume that the sections of cable remain straight during the motion.
   2. Use the methods of section 1.5 and 1.6 for position-dependent force, solve for the cable speed v as a function of the hanging length x.
   3. Further use the methods of section 1.5 and 1.6 to derive an algebraic relation between the hanging length x and time t.
   4. Verify that the solution satisfies both the initial conditions and the relation you obtained for (c). (We will later use differential equation techniques to find this solution much more easily.)

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