Force And Pressure
PRESSURE
A liquid exerts pressure.To show it,consider a vessel containing a liquid.If it has a small hole near the bottom,the liquid will flow out of it.If the motion of the liquid is stopped by pressing a finger against the hole,the finger will experience an outward force due to the liquid contained in the vessek.Therefore,it can be concluded that the liquid exerts pressure on the walls and base of the containing vessel
Thrust.The total force exerted by a liquid on any surface in contact with it ,is called thrust of liquid
Pressure.The thrust exerted by a liquid (at rest) per unit area of the surface in contact with the liquid is called pressure
If F is the thrust exerted by a liquid on a surface of area A,then pressure is given by
\( p = \frac{F} {A} \)
The unit of pressure is dyn c in cgs system and N\( m^{ - 2} \) in SI.
A pressure of 1N\( m^{ - 2} \) is also called1 pascal (Pa)
1Pa (or 1N\( m^{ - 2} \))=10 dyn c\( m^{ - 2} \)
The dimensional formula of pressure is \( [ML^{ - 1} T^{ - 2} ] \)
When a liquid is in equilibrium,forces acting on the liquid must act perpendicular to its surface .It can be easily proved as below:
let us consider a cylindrical vessal of area A and is filled with a liquid of density \( \rho \).To calculate variation of pressure inside of fluid let us consider a small cylindrical liquid element ,in side of a liquid ,of height h.The face area is A(for faces).F1 ,F2 are the forces acting on two faces.the weight of the liquid in small cylindrical element is mg and is at rest because the force acting on top face and bottom face are balanced with each other. here p1 and p2 are pressures on the top and bottom of the cylindrical element.
At equilibrium forcs in vertical direction are balanced with each other
F1 + mg = F2
F1-F2 =mg ( F1 /A = P1 =>F1 = P1 A)( F2/A=P2=> F2=P2A)
(P1 A-P2A) =mg
((P1 -P2)A=mg (m = \( \rho \) V; V= volume of a liquid)
m=\( \rho \)Ah
(P1 -P2)A=\( \rho \)Ahg
(P1 -P2)=\( \rho \)hg
Pressure in a liquid increases with depth.If a vessel contains a liquid,the pressure on the top surface(P1)=\( P_0 \)(atmospheric preesure),the pressure at depth h below the top surface is(P2) P=+\( h\rho g \).Here
is gauge pressure ,P is absolute pressure.
Pressure in a liquid at two points will be the same is they lie at the same horizontal level
Pressure at a point in a liquid depends on the vertical depth and density of liquid only but it is independent of shape of the vessel containing the liquid
The pressure difference between static pressure and atmospheric pressure is known as gauge pressure
P-P0=\( h\rho g \)