Question

A vertical U tube of uniform cross-section contains water in both the arms. A $10cm$ glycerine column (R.D$=1.2$) is added to one of the limbs. The level difference between the free surfaces in the two limbs will be

• A. $4cm$
• B. $2cm$
• C. $6cm$
• D. $8cm$

Answer 1

The correct option is B $2cm$

Let the height of the glycerin in the first arm be $h_h$ = $10cm=0.1m$

Let the height of the water in the second arm be $h_w$

Density of the water = ${\rho }_w$

Density of glycerin = ${\rho }_h$

Let the level difference between the two free surfaces in the two limbs be x

Pressure at both levels is equal, therefore,

$\begin{array}{c}{p}_w=p_h\\ {\rho }_{h}g{h}_h={\rho }_{w}g{h}_w\\ h_w=h_h+x\\ {\rho }_h{h}_h={\rho }_w{h}_h+{\rho }_{w}x\\ x=\frac{{\rho }_h{h}_h-{\rho }_w{h}_h}{{\rho }_w}=\frac{0.1\left(1200\frac{kg}{m^3}-1000\frac{kg}{m^3}\right)}{1000\frac{kg}{m^3}}=0.002m\end{array}$

Therefore the level difference = $2cm$