Ch 3 2 - asdasd PDF

Preview

Summary

asdasd...

Description

3.4 Hydrostatic Forces on Plane Surfaces ¾ If a plane surface immersed in a fluid is horizontal, then  Hydrostatic pressure is uniform over the entire surface.  The resultant force acts at the centroid of the plane. F

¾ If a plane surface immersed in a fluid is not horizontal, then  Hydrostatic pressure is linearly distributed over the surface.  The magnitude and location of the resultant force are obtained by a more general type of analysis. F

¾ Consider the plane surface AB immersed in a liquid and inclined at angle a to the liquid surface as shown in fig. 3.16.  The resultant hydrostatic force on the plane surface is

F = (γ y sin α ) A

(11)

where y is the distance from the liquid surface to the centroid of the plane surface and A is the area of the surface.

 From equation (11) above, we can conclude that the magnitude of the resultant hydrostatic force on a plane surface is the product of the pressure at the centroid of the surface and the area of the surface, i.e. (12) F = pA Note: For most hydrostatic problems one side of the solid surface is exposed to the atmosphere. The forces due to atmospheric pressure on both sides of the surface cancel out. Therefore, the pressure used to calculate the force in equation (12) is gage pressure.

 The point where the resultant force acts on the surface is called the center of pressure (CP).  The vertical location of the center of pressure from the liquid surface can be obtained by finding the slant distance from the liquid surface to CP, which is: y cp = y +

I yA

(13)

where Ī is the second moment of area of the plane surface. See figure A.1 (in the Appendix) for centroids and 2nd moments of area of various surfaces.  The horizontal location of the center of pressure can be obtained by equating moments about an edge of the surface.

3.5 Hydrostatic Forces on Curved Surfaces ¾ For a curved surface, the pressure forces, being normal to the local area element, vary in direction along the surface and thus cannot be added numerically. ¾ The resultant hydrostatic force is computed by considering the free-body diagram of a body of fluid in contact with the curved surface, as illustrated below.

FCB C B

B

Free-body diagram

W

FAC

F A

A B F

A

¾ The steps involved in calculating the horizontal and vertical components of the hydrostatic force F are as follows:  Summation of direction gives

forces

in

the

horizontal

Fx = FAC where FAC is the hydrostatic force on plane surface AC. It acts through the center of pressure of side AC.

 Summation of forces in the vertical direction gives Fy = W + FCB where W is the weight of the fluid (acting through the center of gravity) of the free-body diagram and FCB is the hydrostatic force (acting through the centroid) on the surface CB.  The line of action of Fy is obtained by summing the moments about any convenient axis.  The hydrostatic force on the curved surface is equal and opposite to the force F on the freebody diagram.

Hydrostatic Forces on Curved Surfaces  For a curved surface, the pressure forces, being normal to the local area element, vary in direction along the surface and thus cannot be added numerically.  The resultant hydrostatic force is computed by considering the free-body diagram of a body of fluid in contact with the curv surface, as illus FCB C

B W

FAC A

Free-body diagram

F

B

F Fx

A

Fy

B



A

 The steps involved in calculating the horizontal and vertical components of the hydrostatic force F are as follows: 

Summation of forces in the horizontal direction gives Fx = FAC where FAC is the hydrostatic force on plane surface AC. It acts through the center of pressure of side AC.



Summation of forces in the vertical direction gives Fy = W + FCB where W is the weight of the fluid (acting through the center of gravity) of the freebody diagram and FCB is the hydrostatic force (acting through the centroid) on the surface CB.



The line of action of Fy is obtained by summing the moments about any convenient axis.



The hydrostatic force on the curved surface is equal and opposite to the force F on the free-body diagram.



The resultant force is given as, F  Fx2  Fy2 .



Its angle from the horizontal is given as,   tan 1 Fy Fx  .



The location of the resultant force on the curved surface is such that the line of action of the force passes through the point of intersection of the horizontal force (Fx) and the vertical force (Fy), as shown.

3.6 Buoyancy Principle of Buoyancy ¾ The general principle of buoyancy is expressed in the Archimedes’ principle, which is stated as follows: For an object partially or completely submerged in a fluid, there is a net upward force (buoyant force) equal to the weight of the displaced fluid.

¾ The buoyant force passes through the centroid of the displaced volume (called the center of buoyancy). Its magnitude is

FB = γ VD where

V D is

the displaced volume (see the figures below). D

A

C

D A

C

B

B

Body is completely submerged in the fluid. The displaced volume is equal to the volume of the body, i.e. volume ABCDA.

Body is partially submerged in the fluid. The displaced volume is equal to volume ABCA.

Some remarks about the weight of an object ¾ The weight of an object in a fluid medium refers to the tension in the spring when the object is attached to a spring balance. ¾ The weight (of the object) registered by the spring balance depends on the medium in which it is measured. [See the illustration below.] T = W - FB T=W

W

W

FB

Body in air

Body in water

¾ The weight commonly referred to in daily use is the weight in air.

Hydrometer ¾ The device used for measuring the specific gravity of a liquid is the hydrometer. It utilizes the principle of buoyancy. ¾ The hydrometer is a glass bulb that is weighted on one end to make it float in a vertical position (see fig. 3.22). ¾ On the stem of the hydrometer is a scale which indicates the specific gravity of the liquid in which it is floating.

Example: Problem 3.99 The hydrometer shown sinks 5.3 cm in water (15oC). The bulb displaces 1.0 cm3, and the stem area is 0.1 cm2. Find the weight of the hydrometer.

3.7 Stability of Immersed and Floating Bodies Immersed Bodies • The stability of an immersed body depends on the relative positions of the center of gravity (G) of the body and the center of buoyancy (C) . [See figure 3.23.]  If the center of buoyancy is above the center of gravity, the body is stable.  If the center of gravity is above the center of buoyancy, the body is unstable.  If the center of buoyancy and center of gravity are coincident, the body is neutrally stable.

Floating Bodies ¾ The stability of a floating body depends on the shape of the body and the position in which it is floating.  If the center of gravity is below the center of buoyancy, the body is stable.  If the center of gravity is above the center of buoyancy, the body is stable if the metacentric height is positive. The metacentric height is the distance from G to the point of intersection of the lines of action of the buoyant force before and after heel (see figure 3.24)....