Question

A massless, inextensible string is wrapped around a solid cylinder of mass M; radius R. The string is pulled up vertically to prevent the center of mass from falling down as the string unwinds itself. The work done by the applied force when cylinder accuires angular speed ω₁ is

(1) \(MR^2ω^2_1\)

(2) \(\frac{MR^2ω^2_1}{2}\)

(3) \(\frac{MR^2ω^2_1}{4}\)

(4) \(\frac{2MR^2ω^2_1}{3}\)

Answer 1

 (3) \(\frac{MR^2ω^2_1}{4}\)

Fig shows cylinder with vertical string pulled up by a force F. As C.M of cylinder is at rest; F = Mg. The torque; \(\tau\) ; due to applied force about O is

\(\tau=F.R=MgR\)

The M.I of cylinder about its symmetry axis = I = \(\frac{MR^2}{2}\)

α = The angular accleration of cylinder = \(\frac{\tau}{1}=\frac{2g}{R}\)

Let ω be the total angle describe by cylinder as it accquires angular speed ω₁ . Then