9.20** Problems for Chapter 9 363 Consider a frictionless puck on a horizontal turntable that is rotating counterclockwise with angular velocity 2. (a) Write down Newton's second law for the coordinates x and y of the puck as seen by me standing on the turntable. (Be sure to include the centrifugal and Coriolis forces, but ignore the earth's rotation.) (b) Solve the two equations by the trick of writing n = x+iy and guessing a solution of the form n = e-iat. [In this case-as in the case of critically damped SHM discussed in Section 5.4-you get only one solution this way. The other has the same form (5.43) we found for the second solution in damped SHM.] Write down the general solution. (c) At time t = 0, I push the puck from position r = (xo, 0) with velocity vo = (xo, Uyo) (all as measured by me on the turntable). Show that x(t) = (x + xof) cos t + (vyo + x)t sin t y(t) = (x + Vxot) sin t + (vyo + x)t cos t (9.72) (d) Describe and sketch the behavior of the puck for large values of t. [Hint: When t is large the terms proportional tot dominate (except in the case that both their coefficients are zero). With large, write (9.72) in the form x(t) = t (B₁ cos t + B₂ sin St), with a similar expression for y(t), and use the trick of (5.11) to combine the sine and cosine into a single cosine-or sine, in the case of y(t). By now you can recognize that the path is the same kind of spiral, whatever the initial conditions (with the one exception mentioned).] 9.21 ** When a puck slides on a rotating turntable, as in Problems 9.20 and 9.24, it can come instantaneously to rest. Sketch the shape of the path when this happens and explain. If you did Problem 9.24, comment on the relevance of this result to part (d) of that problem. 9.22** If a negative charge -q (an electron, for example) in an elliptical orbit around a fixed positive charge is subjected to a weak uniform magnetic field B, the effect of B is to make the ellipse precess slowly-an effect known as Larmor precession. To prove this, write down the equation of motion of the negative charge in the field of Q and B. Now rewrite it for a frame rotating with angular velocity 2. [Remember that this changes both d'r/dt² and dr/dt.] Show that by suitable choice of 2 you can arrange that the terms involving ŕ cancel out, but that you are left with one term involving Bx (B x r). If B is weak enough this term can certainly be neglected. Show that in this case the orbit in the rotating frame is an ellipse (or hyperbola). Describe the appearance of the ellipse as seen in the original nonrotating frame. 1.23** Here is an unusual way to solve the two-dimensional isotropic oscillator- the motion of a particle subject to a force -kr. Show that by choosing a suitable rotating reference frame, you can arrange that the centrifugal force exactly cancels the force -kr. Recalling the analogy between the Coriolis and magnetic forces, you should be able to write down the general solution for the motion as seen in the rotating frame. If you write your solution in the complex form of Section 2.7, then you can transform back to the nonrotating frame by multiplying by a suitable rotating complex number. Show that Do problem 9.20 from page 363. However, you can solve part (b) by any method that you like; you don't have to use the 2+ iy trick if you don't want to.