Question

A solid cylinder has a thin string wrapped several times around its circumference. The string is fixed at one end and the cylinder is released. Find the downward acceleration of cylinder and tension in the string.

Answer 1

`mg-T=ma_(c.m.)` (`i`)
`Tr=I_(0)alpha` (`ii`)
`a_(c.m.)=ralpha` (`iii`)
`I_(0)=(1)/(2)mr^(2)` (`iv`)
From (`ii`), (`iii`) and (`iv`)
`Tr=(1)/(2)mr^(2)(a_(c.m.))/(r)impliesT=(1)/(2)ma_(c.m.)`
`mg-T=ma_(c.m.)`
`mg-(1)/(2)ma_(c.m.)=ma_(c.m.)`
`a_(c.m.)=(2g)/(3)`
`T=(1)/(2)ma_(c.m.)=(mg)/(3)`
OR
The body rolls without slipping on vertical surface
`a_(c.m.)=(gsintheta)/(1+(k^(2))/(R^(2)))=(gsin90^(@))/(1+(1)/(2))=(2)/(3)g`