Consider a frictionless puck on a horizontal turntable

QUESTION:

Consider a frictionless puck on a horizontal turntable that is rotating counterclockwise with angular velocity Q. (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 ri = x iy and guessing a solution of the form /I = e-`at. [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 ro = (x0, 0) with velocity v, = (vxo, vyo) (all as measured by me on the turntable). Show that x(t) = (xo vxot) cos Qt (vyo Cbco)t sin Qt y(t) = (x0 + v xot) sin Qt + (v,,0 + Qxo)t cos Qt I (9.72) (d) Describe and sketch the behavior of the puck for large values of t. [Hint: When t is large the terms proportional to t dominate (except in the case that both their coefficients are zero). With t large, write (9.72) in the form x(t) = t (Bi cos Qt B2 sin Qt), 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).]