上浒充通大￥ Shanghai Jiao Tong University 4.5 Hydrostatic Pressure Difference Between Two Points There are several "rules"or comments which directly result from the above equation: Air 1)If you can draw a continuous line Water through the same fluid from point 1 to point 2,then p=p2 if=. For example,consider the oddly shaped container.By this rule,P=p2 and p4=Ps since these points are at the same elevation in the same fluid.However,P2 does not equal P3 even though they are at the same elevation,because one cannot draw a line connecting these points through the same fluid.In fact,p2 is less than p3 since mercury is denser than water
Shanghai Jiao Tong University 4.5 Hydrostatic Pressure Difference Between Two Points There are several "rules" or comments which directly result from the above equation: 1) If you can draw a continuous line through the same fluid from point 1 to point 2, then p1 = p2 if z1 = z2. For example, consider the oddly shaped container. By this rule, p1 = p2 and p4 = p5 since these points are at the same elevation in the same fluid. However, p2 does not equal p3 even though they are at the same elevation, because one cannot draw a line connecting these points through the same fluid. In fact, p2 is less than p3 since mercury is denser than water
上游充通大学 Shanghai Jiao Tong University 4.5 Hydrostatic Pressure Difference Between Two Points 2)Any free surface open to the atmosphere has atmospheric pressure,po. (This rule holds not only for hydrostatics, but for any free surface exposed to the atmosphere,whether the surface is moving,stationary,flat,or mildly curved.) 2 Consider the hydrostatics example of a container of water: The little upside-down triangle indicates a free surface,and means that the pressure there is atmospheric pressure,po. In other words,in this example,p=po.To find the pressure at point 2,our hydrostatics equation is used:p2 po+pgh (absolute pressure)or p2 pgh(gauge pressure)
Shanghai Jiao Tong University 4.5 Hydrostatic Pressure Difference Between Two Points 2) Any free surface open to the atmosphere has atmospheric pressure, p0. (This rule holds not only for hydrostatics, but for any free surface exposed to the atmosphere, whether the surface is moving, stationary, flat, or mildly curved.) Consider the hydrostatics example of a container of water: The little upside-down triangle indicates a free surface, and means that the pressure there is atmospheric pressure, p0. In other words, in this example, p1 = p0. To find the pressure at point 2, our hydrostatics equation is used: p2 = p0+ρgh (absolute pressure) or p2 = ρgh (gauge pressure)
上游充通大学 Shanghai Jiao Tong University 4.5 Hydrostatic Pressure Difference Between Two Points 3)The shape of a container does not matter in hydrostatics. Jug Graduated Cylinder Vase (Except of course for very small diameter tubes,where surface tension becomes important.) Consider the three containers in the figure below: 2 At first glance,it may seem that the pressure at point 3 would be greater than that at point 1 or 2,since the weight of the water is more "concentrated"on the small area at the bottom,but in reality,all three pressures are identical.Use of our hydrostatics equation confirms this conclusion,i.e. Ppelow=Pabove+PgA→P,=P2=p3=p,+Pg△
Shanghai Jiao Tong University 4.5 Hydrostatic Pressure Difference Between Two Points 3) The shape of a container does not matter in hydrostatics. (Except of course for very small diameter tubes, where surface tension becomes important.) Consider the three containers in the figure below: At first glance, it may seem that the pressure at point 3 would be greater than that at point 1 or 2, since the weight of the water is more "concentrated" on the small area at the bottom, but in reality, all three pressures are identical. Use of our hydrostatics equation confirms this conclusion, i.e. below above 1 2 3 0 p p g z p p p p gz = + Δ⇒ = = = + Δ ρ ρ
上游充通大学 4.5 Hydrostatic Pressure Difference Between Two Points Shanghai Jiao Tong University 4) Pressure in layered fluid. For example,consider the container in the figure below,which is partially filled with mercury,and partially with water: P=0 YH.o H20 p=PwatergAz Hg P2 Pmercury gAz2 p
Shanghai Jiao Tong University 4.5 Hydrostatic Pressure Difference Between Two Points 4) Pressure in layered fluid. For example, consider the container in the figure below, which is partially filled with mercury, and partially with water: 1 water 1 p = Δ ρ g z 2 mercury 2 1 p g = ρ Δ + z p 0 p = 0 H O2 γ Hg γ 1 1
上游充通大学 Shanghai Jiao Tong University 4.5 Hydrostatic Pressure Difference Between Two Points In this case,our hydrostatics equation must be used twice,once in each of the liquids Pucow =Pabove+pgA →p1=Po+Owater8A1 and p2=p+Pmercury gAZ Combining, P2=Po+Pater gA+Pmercuy gA2 Shown on the right side of the above figure is the distribution of pressure with depth across the two layers of fluids,where the atmospheric pressure is taken to be zero po =0. The pressure is continuous at the interface between water and mercury.Therefore,Pi,which is the pressure at the bottom of the water column,is the starting pressure at the top of the mercury column
Shanghai Jiao Tong University 4.5 Hydrostatic Pressure Difference Between Two Points In this case, our hydrostatics equation must be used twice, once in each of the liquids below above 1 0 water 1 2 1 mercury 2 2 and Combining, p p gz p p g z pp g z p p ρ ρ ρ = +Δ ⇒=+ Δ =+ Δ = 0 water 1 mercury 2 + Δ+ Δ ρ ρ gz gz Shown on the right side of the above figure is the distribution of pressure with depth across the two layers of fluids, where the atmospheric pressure is taken to be zero . 0 p = 0 The pressure is continuous at the interface between water and mercury. Therefore, , which is the pressure at the bottom 1 p of the water column, is the starting pressure at the top of the mercury column