A thin plate `AB` of large area `A` is placed symmetrically in a small gap of height `h` filled with water of viscosity `eta_(0)`and the plate has a constant velocity `v` by applying a force `F` as shown in the figure. If the gap is filled with some other liquid of viscosity `0.75 eta_(0)` at what minimum distance (in cm) from top wall should the plate be placed in the gap, so that the plate can again be pulled at the same constant velocity `V`. by applying the same force `F`? (Take `h=20 cm`)

Answer

5

Explanation

Case I `Y_(1)=Y_(2)=(h)/(2),(dv)/(dx)=(v)/(h//2)=(2v)/(h)" "Rightarrow" ""Viscous force"=eta_(0)A(dv)/(dx)=eta_(0)A(2v)/(h)`
As plates have 2 sides, Total force = `(4eta_(0)vA)/(h)`
Case II `F=(etavA)/(y)+(etavA)/(h-y)=(etavAh)/(y(h-y))`
This force should be equal as in case I
`(etavAh)/(y(h-y))=(4eta_(0)vA)/(h)Rightarrow(3)/(16)h^(2)=y(h-y)Rightarrowy=(h)/(4),(3h)/(4)`
For minimum distance from top, `y=(h)/(4)=(20)/(4)=5cm`