Hydrostatics

PASCAL'S LAW

Pressure difference (P – P_a) = hpg. This is Pascal's Low which states that Hydraulic lift is common application of Pascal's Law.

(i) Hydraulic press.

$p=\frac{f}{a}=\frac{F}{A}orF=\frac{A}{a}\times f$

This can be used to lift a heavy load placed on the platform of larger piston or to press the things placed between the piston and the heavy platform.

(ii) Hydraulic Brake.

Hydraulic brake system is used in auto-mobiles to retard the motion.

HYDROSTATIC PARADOX

Pressure is directly proportional to depth and by applying Pascal's law it can be seen that pressure is independent of the size and shape of the containing vessel.

$P_A=P_B=P_C$

ATMOSPHERIC PRESSURE

Definition: The atmospheric pressure at any point is numerically equal to the weight of a column of air of unit cross-sectional area extending from that point to the top of the atmosphere.

MEASUREMENT OF ATMOSPHERIC PRESSURE

(i) Mercury Barometer

To measure the atmospheric pressure experimentally, Torricelli invented a mercury barometer in 1643.

$p_a=h\rho g$

The pressure exerted by a mercury column of 1mm high is called 1 Torr.

1 Torr = 1 mm of mercury column

(ii) Water Barometer.

Let us suppose water is used in the barometer instead of mercury.

$h\rho g=1.013\times {10}^{5}orh=\frac{1.013\times {10}^5}{\rho g}$

The height of the water column in the tube wilt be 10.3 m.

ARCHIMEDES' PRINCIPLE

According to this principle, when a body is immersed wholly or partially in a fluid, it loses its weight which is equal to the weight of the fluid displaced by the body.

Up thrust = buoyancy = $V{\rho }_{\ell }g$ V = volume submerged ${\rho }_{\ell }$= density of liquid.

Relation between density of solid and liquid

weight of the floating solid = weight of the liquid displaced

$V_1{\rho }_{1}g=V_2{\rho }_{2}g\Rightarrow \frac{{\rho }_1}{{\rho }_2}=\frac{V_2}{V_1}$

or $\frac{\text{Density of solid}}{\text{Density of liquid}}=\frac{\text{Volume of the immeresed portion of the solid}}{\text{Total Volume of the solid}}$

This relationship is valid in accelerating fluid also.

PRESSURE IN CASE OF ACCELERATING FLUID

(i) Liquid Placed in elevator: When elevator accelerates upward with acceleration $a_0$ then pressure in the fluid, at depth 'h' may be given by,

$p=h\rho \left[g+a_0\right]$

and force of buoyancy,$B=m\left(g+a_0\right)$

(ii) Free surface of liquid in horizontal acceleration:

$\tan \theta =\frac{a_0}{g}$

$p_1-p_2=\ell \rho a_0$ where $p_1$and $p_2$ are pressures at point 1 & 2. Then $h_1-h_2=\frac{\ell a_0}{g}$

(iii) Free surface of liquid in case of rotating cylinder.

$h=\frac{v^2}{2g}=\frac{{\omega }^2{r}^2}{2g}$