Question

A thread is wound around cylinder of mass $M$ and radius $R$. It is allowed to fall as shown. Find its acceleration.

• A. $\frac{g}{3}$
• B. $\frac{4g}{3}$
• C. $\frac{2g}{3}$
• D. None

Answer 1

The correct option is C $\frac{2g}{3}$




The spool undergoes translation as well as rotation.
For translation,
$Mg-T=Ma\dots \dots \left(1\right)$
For rotation, about $C$
$I_C=\frac{M{R}^2}{2}$
$\tau =T.R$
As $\tau =I\alpha$
$T.R=\frac{M{R}^2}{2}.\alpha \dots \dots \left(2\right)$
Constraint equation
${\overrightarrow{a}}_{PC}=\alpha R\uparrow$
${\overrightarrow{a}}_{CP}=\alpha R\downarrow$
${\overrightarrow{a}}_{CG}={\overrightarrow{a}}_{CP}+{\overrightarrow{a}}_{PG}$
${\overrightarrow{a}}_{PG}=0$
(As point $P$ and $G$ are two points on an inextensible thread).
$\Rightarrow {\overrightarrow{a}}_{CG}={\overrightarrow{a}}_{CP}=\alpha R\downarrow$
or $a=\alpha R\dots \dots \left(3\right)$
Use (3) in (2)
$T=\left(\frac{MR}{2}\right)\frac{a}{R}=\frac{Ma}{2}$
Put this in (1),
$Mg-\frac{Ma}{2}=Ma$
$Mg=\frac{3Ma}{2}\Rightarrow a=\frac{2g}{3}$