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30. Problem 19.17 (Fey) A yo-yo like spool consists of two uniform disks, each of mass
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and radius R, and an axle of radius r and negligible mass. A thread wound around
the axle is attached to the ceiling and the spool is released from rest a
distance below the ceiling. We have to find the downward acceleration of the
centre of the spool.
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Solution: Click For PDF Version Moment of inertia of each disk is
Axle of the Yo-yo disks rolls down on release without sliding. Therefore,
there are two types of accelerations of the yo-yo, the translational
acceleration of the centre of mass,
From the free-body diagram we note that the net force acting vertically
downwards is
and
Also,
By eliminating
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and as the
axle has negligible mass, the moment of inertia of the yo-yo will be
.
, and the
angular acceleration,
, of the
yo-yo about its axis. Application of the Newton’s second law of motion and its
counterpart for rotational motion gives two algebraic equations. Let T be the
tension in the string. The free-body diagram of the yo-yo is 
and the
torque,
, which
results in angular acceleration in the anticlockwise direction. We get two
equations of motion
,
.
.
, we get
.