Question

A cylindrical container of radius $R$ and height $h$ is completely filled with a liquid. Two horizontal $L-$shaped pipes of small cross-sectional area $a$ are connected to the cylinder as shown in the figure. Now the two pipes are opened and fluid starts coming out of the pipes horizontally in opposite directions. Then the torque due to ejected liquid on the system is

• A. $\text{4 a g h}\rho R$
• B. $\text{8 a g h}\rho R$
• C. $\text{2 a g h}\rho R$
• D. $\text{3 a g h}\rho R$

Answer 1

The correct option is A $\text{4 a g h}\rho R$

Velocity of efflux of water $\left(v\right)=\sqrt{2g\left(\frac{h}{2}\right)}=\sqrt{gh}$

Force on ejected water=rate of charge of momentum of ejected water

$=\rho \left(av\right)\left(v\right)$

$=\rho {\left(av\right)}^2$

Torque of this force about central line

$=\left(\rho a{v}^2\right)2R.2$

$=4\rho a{v}^{2}R=4\rho aghR(\because v^2=gh)$