. In part (B) of Example 10.2, the compact disc was modeled as a rigid object under constant angular acceleration to find the total angular displacement during the playing time of the disc. In reality, the angular acceleration of a disc is not constant. In this problem, let us explore the actual time dependence of the angular acceleration. (a) Assume the track on the disc is a spiral such that adjacent loops of the track are separated by a small distance h. Show that the radius r of a given portion of the track is given by where ri is the radius of the innermost portion of the track and u is the angle through which the disc turns to arrive at the location of the track of radius r. (b) Show that the rate of change of the angle u is given by where v is the constant speed with which the disc surface passes the laser. (c) From the result in part (b), use integration to find an expression for the angle u as a function of time.
. In part (B) of Example 10.2, the compact disc was modeled as a rigid object under constant
to find the total
time of the disc. In reality, the angular acceleration of a
disc is not constant. In this problem, let us explore the
actual time dependence of the angular acceleration.
(a) Assume the track on the disc is a spiral such that adjacent loops of the track are separated by a small distance h.
Show that the radius r of a given portion of the track is
given by
where ri is the radius of the innermost portion of the
track and u is the angle through which the disc turns to
arrive at the location of the track of radius r. (b) Show
that the rate of change of the angle u is given by
where v is the constant speed with which the disc surface
passes the laser. (c) From the result in part (b), use integration to find an expression for the angle u as a function
of time. (d) From the result in part (c), use differentiation to find the angular acceleration of the disc as a function of time.
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