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 b/w the two free surface in the two limbs will be ????????????????????????????????????????

Answer 1

A vertical U tube of uniform cross-section contains water in both the arms. A 10cm glycerin column (R.D=1.2) is added to one of the limbs. The level difference b/w the two free surface in the two limbs will be?

**Solution:**

$h_h = 10 \, \text{cm} = 0.1 \, \text{m}$ - hight of the glycerin in first arm;

$h_w$ - hight of the water in second arm;

$\rho_w = 1000 \frac{\mathrm{kg}}{\mathrm{m}^3}$ - density of the water;

$\rho_h = 1200 \frac{\mathrm{kg}}{\mathrm{m}^3}$ - density of the glycerin;

x - level difference b/w the two free surface in the two limbs;

Pressures at level AA' of the water and glycerin are equal:

$AA^{\prime}:p_w = p_h$

$\rho_hgh_h = \rho_wgh_w$ (1)

$h_w = h_h + x$ (2)

(2)in(1):

$\rho_hgh_h = \rho_wg(h_h + x)$

$\rho_h h_h = \rho_w h_h + \rho_w x$

$$x = \frac {\rho_ {h} h _ {h} - \rho_ {w} h _ {h}}{\rho_ {w}} = \frac {0 . 1 m \left(1 2 0 0 \frac {k g}{m ^ {3}} - 1 0 0 0 \frac {k g}{m ^ {3}}\right)}{1 0 0 0 \frac {k g}{m ^ {3}}} = 0. 0 2 m = 2 c m$$

**Answer:** the level difference b/w the two free surface in the two limbs will be 2cm.