Question

If the velocity of water layers in river is given by $v=k{y}^2$, where $y$ is the height from the river bed. The force required to keep a square plate of area $A$ with constant velocity at height $h$ is $[\eta$ = coefficient of viscosity]

• A. $\eta Ak\frac{h}{2}$
• B. $4\eta Akh$
• C. $2\eta Akh$
• D. $\eta Akh$

Answer 1

The correct option is C $2\eta Akh$

Since $v=k{y}^2$
$\frac{dv}{dy}=2ky$

The viscous drag is,
$F_r=\eta A\frac{dv}{dy}$
$F_r=2\eta Aky.$

For the topmost layer, $y=h$
$F_{\eta }=2\eta Akh$

The applied force $F$ must balance $F_{\eta }$ since the velocity is constant.
$F=F_{\eta }=2\eta Akh.$