Lecture no 11 & 12 HYDROSTATIC FORCE AND PRESSURE ON PLATES Prepared by Engr.Sarafaraz Khan Turk Lecturer at IBT LUMHS Jamshoro

HYDROSTATIC FORCE AND PRESSURE Fluid statics or hydrostatics is the branch of fluid mechanics that studies fluids at rest. It embraces the study of the conditions under which fluids are at rest in stable equilibrium; and is contrasted with fluid dynamics, the study of fluids in motion

HYDROSTATIC FORCE AND PRESSURE The hydrostatic pressure can be determined from a control volume analysis of an infinitesimally small cube of fluid. Since pressure is defined as the force exerted on a test area (p = F/A, with p: pressure, F: force normal to area A, A: area), and the only force acting on any such small cube of fluid is the weight of the fluid column above it, hydrostatic pressure can be calculated according to the following formula:

HYDROSTATIC FORCE AND PRESSURE where: p is the hydrostatic pressure (Pa), ρ is the fluid density (kg/m3), g is gravitational acceleration (m/s2), A is the test area (m2), z is the height (parallel to the direction of gravity) of the test area (m), z0 is the height of the zero reference point of the pressure (m).

Scuba Diving and Hydrostatic Pressure

HYDROSTATIC FORCE AND PRESSURE Deep-sea divers realize that water pressure increases as they dive deeper. This is because the weight of the water above them increases.

Scuba Diving and Hydrostatic Pressure Pressure on diver at 100 ft? Danger of emergency ascent? 1 100 ft 2 Boyle’s law If you hold your breath on ascent, your lung volume would increase by a factor of 4, which would result in embolism and/or death.

HYDROSTATIC FORCE AND PRESSURE Suppose that a thin plate with area A m2 is submerged in a fluid of density ρ kg/m3 at a depth d meters below the surface of the fluid.

HYDROSTATIC FORCE AND PRESSURE The fluid directly above the plate has volume V = Ad So, its mass is: m = ρV = ρAd

Thus, the force exerted by the fluid on the plate is F = mg = ρgAd HYDROSTATIC FORCE Thus, the force exerted by the fluid on the plate is F = mg = ρgAd where g is the acceleration due to gravity.

The pressure P on the plate is defined to be the force per unit area: HYDROSTATIC PRESSURE The pressure P on the plate is defined to be the force per unit area:

HYDROSTATIC PRESSURE The SI unit for measuring pressure is newtons per square meter—which is called a pascal (abbreviation: 1 N/m2 = 1 Pa). As this is a small unit, the kilopascal (kPa) is often used.

HYDROSTATIC PRESSURE For instance, since the density of water is ρ = 1000 kg/m3, the pressure at the bottom of a swimming pool 2 m deep is:

HYDROSTATIC PRESSURE An important principle of fluid pressure is the experimentally verified fact that, at any point in a liquid, the pressure is the same in all directions. This is why a diver feels the same pressure on nose and both ears.

HYDROSTATIC PRESSURE Equation 1 Thus, the pressure in any direction at a depth d in a fluid with mass density ρ is given by:

HYDROSTATIC FORCE AND PRESSURE This helps us determine the hydrostatic force against a vertical plate or wall or dam in a fluid. This is not a straightforward problem. The pressure is not constant, but increases as the depth increases.

A dam has the shape of the trapezoid shown below. HYDROSTATIC F AND P Example 1 A dam has the shape of the trapezoid shown below. The height is 20 m. The width is 50 m at the top and 30 m at the bottom.

HYDROSTATIC F AND P Example 1 Find the force on the dam due to hydrostatic pressure if the water level is 4 m from the top of the dam.

We choose a vertical x-axis with origin at the surface of the water. HYDROSTATIC F AND P Example 1 We choose a vertical x-axis with origin at the surface of the water.

The depth of the water is 16 m. HYDROSTATIC F AND P Example 1 The depth of the water is 16 m. So, we divide the interval [0, 16] into subintervals of equal length with endpoints xi. We choose xi* [xi–1, xi].

HYDROSTATIC F AND P Example 1 The i th horizontal strip of the dam is approximated by a rectangle with height Δx and width wi

From similar triangles, HYDROSTATIC F AND P Example 1 From similar triangles,

HYDROSTATIC F AND P Example 1 Hence,

If Ai is the area of the strip, then HYDROSTATIC F AND P Example 1 If Ai is the area of the strip, then If Δx is small, then the pressure Pi on the i th strip is almost constant, and we can use Equation 1 to write:

HYDROSTATIC F AND P Example 1 The hydrostatic force Fi acting on the i th strip is the product of the pressure and the area:

HYDROSTATIC F AND P Example 1 Adding these forces and taking the limit as n → ∞, the total hydrostatic force on the dam is: