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Cylinder A is moving downward with a velocity of 9 ft/s when the brake is suddenly applied to the drum. Knowing that the cylinder moves 18 ft downward before coming to rest and assuming uniformly accelerated motion, determine (a) the angular acceleration of the drum, (b) the time required for the cylinder to come to rest.

Step-by-step

Block A:{ v }^{ 2 }-{ v }_{ 0 }^{ 2 }=2as

0-{ \left( 9ft/s \right) }^{ 2 }=2a\left( 18ft \right)

a=-2.25ft/{ s }^{ 2 }\quad ,a=2.25ft/{ s }^{ 2 }\uparrow

Drum:{ v }_{ A }=r{ \omega }_{ 0 }

9ft/s=\left( 0.75ft \right) \omega

{ \omega }_{ 0 }=12rad/s

(a) \quad a=r\alpha

-\left( 2.25ft/{ s }^{ 2 } \right) =\left( 0.75ft \right) \alpha \\

\alpha =-3\quad rad/{ s }^{ 2 }\quad ,\alpha =3.00rad/{ s }^{ 2 }\curvearrowright

(b) Uniformly accelerated motion.\omega =0\quad when \quad t={ t }_{ 1 }

\omega ={ \omega }_{ 0 }+\alpha t:\quad 0=\left( 12rad/s \right) -\left( 3rad/{ s }^{ 2 } \right) { t }_{ 1 }\quad \quad ,{ t }_{ 1 }=4.00\quad s

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