Derivation Of Pascal's Law, Part 1 (1943)

[Music] the study of naval ordnance and fire control is constantly making more necessary a knowledge of the operation of hydraulic mechanisms hydraulic equipment is responsible for the ease with which we are able to train 16-inch turrets for the automatic operation of the guns our naval aircraft and for the smooth action of our rapid-fire anti-aircraft guns most all naval ordnance equipment depends to a large extent an electric hydraulic power transmission for its operation as a result a knowledge of the working principles of hydraulics is necessary in order to operate maintain and repair the modern ordnance equipment which is today in wide use throughout the fleet hydraulics deals with the properties and behavior of liquids at the outset we will demonstrate a few simple principles which when thoroughly understood will enable us to figure out the workings of complex hydraulic systems and machines if you were asked to demonstrate that one liquid were heavier than another you would find that you could go about it in only one way that way would be to take equal amounts of the liquids and to weigh them let us take water and oil as examples we are to compare the heaviness of these two liquids for convenience we will take a cubic foot of each and weigh it we take the water first if we place a cubic foot of water on a scale which has been set to compensate for the weight of the container in other words so that the weight of the empty container does not register on the scale we will find that the cubic foot of water weighs sixty two point four pounds now let us weigh the oil we place the container of oil on the scale and find the it weighs 50 pounds thus by weighing a cubic foot of water and the same amount of oil we have found that the water is heavier than the oil if we place the containers of water and oil on opposite sides of the scale we see readily that the water is heavier than the equal amount of oil we can illustrate this even further by pouring oil on water when we do so we find that the oil will float this again indicates that oil is the lighter of the two substances remember that in comparing the weight of the two liquids we used one cubic foot of each this is referred to as a unit volume the same would be true had we used one cubic inch of each liquid the weight of a unit volume of a liquid we call its density we found that a cubic foot of water weighed sixty two point four pounds thus its density is 62 point four pounds per cubic foot in the case of the oil we found that a cubic foot weighed 50 pounds thus the density of this oil is 50 pounds per cubic foot the density of any liquid may consequently be defined as its weight per unit volume to find the density of any liquid whose weight and volume No we can use the following formula density equals weight divided by volume in the Navy weight is most commonly measured in pounds and volume is usually measured in cubic inches or cubic feet thus the units in which we most commonly measure density are pounds per cubic foot or pounds per cubic inch we have all used the term pressure when referring to a certain force exerted by a liquid actually pressure and force are two different things and should not be confused let us consider a container of liquid as we have just demonstrated in the case of density this liquid has weight the weight of this liquid presses against the bottom and sides of the container holding it let us imagine for a moment the weight of a portion of this liquid as it presses downward against the bottom of the container let us consider a square column of the liquid with a cross section of one square inch and a height equal to the depth of the liquid in the container if it were possible as of course it is not to take this column of liquid out and weigh it we might find that it weighed two pounds this means that the liquid presses down on a scale with a force equal to two pounds if this is so then in the container this square column of liquid which remember is one square inch in cross-section must press down on a square inch of the container bottom with a weight of two pounds expressing this in another way it can be seen that the liquid exerts a force of two pounds on a single square inch of the bottom of the container if you have followed this illustration thus far you have done the kind of thinking that you will have to do in considering pressure for pressure is the force exerted by a liquid on a unit area don't let the words unit area frighten you this simply means a single or unit portion of the area in this case one square inch of the area of the bottom of the container thus we define the pressure as the force a liquid exerts on a unit of the area on which it rests or presses more simply pressure is the force per unit area expressed in a simple formula pressure equals force divided by area the force of the liquid was expressed in pounds the area on which the liquid rests is expressed in square inches thus we express pressure in pounds per square inch in the example we have just used we found that the liquid in the square column exerted a force of two pounds on one square inch at the bottom of the container thus the pressure of the liquid is two pounds per square inch since pressure is the force of a liquid on a unit of the area it does not vary or change with the shape of the vessel containing the liquid for example let us take the container we have just been examining and enlarge the bottom so that it will hold more water we will keep the liquid in the container at the same height if we consider only the liquid pressing down on one square inch of the bottom of this enlarged container we find that the force the liquid exerts on this square inch is the same as it was in the original container for as long as the height of the liquid is the same the force of the liquid on a square inch of the bottom surface has nothing to do with the size of the container or how large the bottom surface is in the same fashion let us make the bottom of our original container smaller and slope the sides inward again the height of the liquid is the same and again the force on a single square inch of the bottom is the same the fact that there is less water in this container then there was in our original one makes no difference since the first of the liquid on a square inch of the bottom is the pressure we can say that the pressure is the same for all three containers in other words the pressure exerted by the liquid is independent of the shape of the vessel let us go back again to our original container we have shown that the pressure on the bottom of the container is 2 pounds per square inch which means that on every square inch of the bottom the liquid exerts a force of 2 pounds if the liquid exerts a force of 2 pounds on a single square inch of the bottom of the container on 2 square inches it would exert 4 pounds in the same manner the force on 3 square inches would be 6 pounds thus we can see that if we wish to find the total force on the bottom of the container we can see that we would multiply the force on one square inch of the bottom by the number of square inches in the bottom of the container we can show this in another way by considering the bottom of the container by itself suppose we're looking up at the bottom on which the liquid rests we see again that the force on one square inch is 2 pounds we want to find the total force on the bottom of the container to do this we must multiply the force on one square inch by the total number of square inches in the bottom and again we see that the total force on the bottom is equal to the force on one square inch times the number of square inches in the bottom we can express this formula more simply because the force on one square inch is we know the pressure and the total number of square inches in the bottom is the area of the bottom thus we now have a simple formula for finding the total force of a liquid on any horizontal surface it is the pressure of the liquid times the area of the surface this formula shows the difference between the terms pressure and force as we use them in hydraulics total force is expressed in pounds we may see why by investigating the units for the right-hand side of the formula pressure is expressed in pounds per square inch and area in square inches we can multiply these units exactly as we would multiply figures if we do so we find that the square inch-- terms cancel out and leave us with pounds as the only dimension thus total force is expressed in pounds applying this formula to the container we have been using the pressure of the liquid was 2 pounds per square inch if the area of the bottom is 20 square inches the total force is the pressure 2 pounds per square inch times the area 20 square inches thus the total force on the bottom of the container is 30 pounds a moment ago we found the pressure acting on the bottom of the container it is possible to find the pressure at any point within the liquid for example suppose we were to find the pressure halfway down to do this we would imagine a square column of liquid half as deep as our previous one this column would obviously weigh half as much as the one which extended all the way down to the bottom consequently the pressure halfway down would be one pound per square inch this shows us that the pressure of a liquid varies with its depth and increases as the depth is increased in other words as the liquid gets deeper the pressure on the bottom is greater this after all is only common sense for example if we were diving we would find that there was a greater weight of water pushing on us when we were on the bottom of the ocean then when we were just a few feet below the surface it may help to remember density and pressure if you will bear in mind that they are both defined in terms of units density of a liquid is its weight per unit volume pressure is the force a liquid exerts on a unit area of surface while the total force on a surface is the pressure of the liquid times the path of the surface it plows now for a review of the principles we have just covered