A cylinder of mass m is suspended through two strings wrapped around it as shwon in figure . Find (a). the tension T in the string and (b). the speed o the cylinder as it falls through a distance h.

Explanation

For the linear motion of the centre of mass.
Net force `= mg-2T=ma`,
[where a is the acceleration of the CM] `" "` ….(i)
For the rotational motion about the CM, Net torque `= 2Tr = l_(cm)(a)/(r )`
or `2T=l_(cm)(a)/(r^(2)) " "` ...(ii)
From (i) and (ii), we get
`a=(mg)/(m+(l_(cm))/(r^(2))) " "` ....(A)
From (ii), `T=(l_(cm))/(2r^(2))((mg)/(m+(l_(cm))/(r^(2)))) " "` ....(B)
Putting `l_(cm)=(1)/(2)mr^(2)`, we get `a=(2)/(3)g` and `T=(mg)/(6)`
`v^(2)=2ah =2((2)/(3)g)h` or `v=sqrt((4gh)/(3))`