First slide

Rotational motion

Question

A string is wrapped around a cylinder of mass M and radius R. The string is pulled vertically upwards to prevent the centre of mass from falling as the cylinder unwinds the string. The tension in the string is

Moderate

A

$\frac{M g}{6}$
B

$\frac{M g}{3}$
C

$\frac{M g}{2}$
D

$\frac{2 M g}{3}$

Solution

Equation of motion is
$M g - T = M a . . . . . . . . . . (i i)$
Taking torque about the axis passing through the centre of the spool and perpendicular to it,
$\begin{array}{l} T R = I α = \frac{1}{2} M R^{2} (\frac{a}{2}) \\ T = \frac{1}{2} M a . . . . . . . . . . . . (i i) \end{array}$
From Eqs. (i) and (ii),
$\begin{array}{l} M a = M g - \frac{1}{2} M a \\ a = \frac{2 g}{3} \\ ∴ T = \frac{M g}{3} \end{array}$

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