Compound-Wheel Cascade

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25.Compound-Wheel Cascade

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Identical spools, each with a hub of radius

r

and rims of radius

3r

at its ends sharing a common axis, are stacked in two layers. The lower row of spools roll without slipping on a fixed floor at their hubs and support a plate on their rims. A second row of spools sits on that plate with their hubs rolling without slipping on it, supporting a second plate on their rims. All spools rotate with the same angular speed

\omega

in the same sense. What is the speed of the upper plate?

$Side view of the stack, drawn with two cylinder layers between three horizontal lines (floor, middle plate, top plate). Each spool is shown as a cross-section: a small hub circle of radius $r$ tangent to the surface below, concentric with a larger rim circle of radius $3r$ tangent to the surface above. Show three spools in the bottom row resting on the floor (hubs touching the floor) with their rims supporting the middle plate, and three more spools in the top row with hubs on the middle plate and rims supporting the top plate. On one bottom-row spool, label the hub radius $r$ and the rim radius $3r$. On one spool in each row, draw a curved arrow indicating rotation direction with the label $\omega$ (same sense in both rows).$

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