Question

In the following figure, a body of mass m is tied at one end of a light string and this string is wrapped around the solid cylinder of mass M and radius R. At the moment t = 0 the system starts moving. If the friction is negligible, angular velocity at time $t$ would be

• A. $\frac{mgRt}{(M + m)}$
• B. $\frac{2Mgt}{(M + 2m)}$
• C. $\frac{2mgt}{R(M - 2m)}$
• D. $\frac{2mgt}{R(M + 2m)}$

Answer 1

Answer: (D)

$\frac{2mgt}{R(M + 2m)}$

Answer 2

We know the tangential acceleration $a = \frac{g}{1 + \frac{I}{mR^{2}}}$

$= \frac{g}{1 + \frac{1/2MR^{2}}{mR^{2}}} = \frac{2mg}{2m + M}$ [As$I = \frac{1}{2}MR^{2}$ for cylinder]

After time t, linear velocity of mass m, $v = u + at$ $= 0 + \frac{2mgt}{2m + M}$

So angular velocity of the cylinder $\omega = \frac { v } { R } =$ $\frac{2mgt}{R(M + 2m)}$

or $\omega = 3v/2L$