Two weights $w_1$ and $w_2$ are suspended from the ends of a light string passing over a smooth fixed pulley. If the pulley is pulled up at an acceleration $g$, the tension in the string will be

The correct option is D $\frac{4w_1{w}_2}{w_1+w_2}$
Given: Acceleration of pulley, $a_p=g$
Let us consider the weight $w_2$ is going downward with an acceleration $a$.
The effective acceleration due to gravity acting is given by,

$g_{eff}=g+g=2g$

The weight of both the block changes to $2w_1$ and $2w_2$
Equating all the forces for $m_1$ vertically, we get,

$T-2w_1=\frac{w_1}{g}a$ .....$\left(1\right)$

Similarly for $m_2,$ Resolving all the forces,we get,

$2w_2-T=\frac{w_2}{g}a$ .....$\left(2\right)$

Dividing $\left(1\right)$ by $\left(2\right)$, we get,
$\frac{w_1}{w_2}=\frac{T-2w_1}{2w_2-T}$

$\Rightarrow 2w_1{w}_2-w_{1}T=w_{2}T-2w_1{w}_2$

$\Rightarrow 4w_1{w}_2=T(w_1+w_2)$

$\Rightarrow T=\frac{4w_1{w}_2}{w_1+w_2}$
Hence, option $\left(A\right)$ is correct