Question

A string is wrapped several times round a solid cylinder and then the end of the string is held stationary while the cylinder is released from rest with an initial motion. The acceleration of the cylinder and tension in the string will be:
\[\begin{align}
  & A.\dfrac{2g}{3}and\dfrac{mg}{3} \\
 & B.gand\dfrac{mg}{2} \\
 & C.\dfrac{g}{3}and\dfrac{mg}{2} \\
 & A.\dfrac{g}{2}and\dfrac{mg}{3} \\
\end{align}\]

Answer 1

Hint: To find the tension is the string, we need to calculate the net force of the string. Since the cylinder undergoes rotation when the string is pulled, we can find its torque which in turn is used to calculate the acceleration of the cylinder.
Formula: $\tau=f\times d$, $\tau=I\times \omega$ and $I=\dfrac{1}{2}mr^{2}$

Complete answer:
Let the mass of the given cylinder which is wound up by the strings be $m$, and $r$ its radius.
Then the moment of inertia experienced by the cylinder during a rotation is given by $I=\dfrac{1}{2}mr^{2}$
Then the torque experienced by the cylinder is given as $\tau=I\times \omega$ where $\omega$ is the angular acceleration. We know that the angular acceleration is given as $\omega=\dfrac{a}{r}$ where $a$ is the linear acceleration and $r$ is the radius of the cylinder.
 Also the torque is given as $\tau=f\times d$ where $f$ is the force acting on the cylinder at a perpendicular distance $d$. Here $d=r$ and $f=T$ where $T$ is the tension on the strings which are wound around the cylinder.
Substituting the values we get, $T\times r=\dfrac{1}{2}mr^{2}\times \dfrac{a}{r}$
Or, $T=\dfrac{1}{2}ma$
Also $F_{net}=W-T$ where $W=mg$ is the weight of the cylinder.
Then, $F_{net}=mg-\dfrac{1}{2}ma$
We know that the net force is given as $F_{net}=ma$ .
Then, $ma=mg-\dfrac{1}{2}ma$
$\Rightarrow mg=\dfrac{3}{2}ma$
$\Rightarrow a=\dfrac{2g}{3}$
Substituting the value of $a$ in $T$ we get, $T=\dfrac{1}{2}ma=\dfrac{mg}{3}$.
Hence we get $a=\dfrac{2g}{3}$ and $T=\dfrac{mg}{3}$

Thus the answer is \[A.\dfrac{2g}{3}and\dfrac{mg}{3}\].

Note:
Here, we are assuming the wound up strings as a solid cylinder with some mass and radius. This is used to simplify the problem and helps in the easy visualisation of the problem. Here, we consider the angular acceleration of the cylinder in terms of the linear acceleration as the string undergoes linear acceleration.