What does pressure equal

Another common type of pressure is that exerted by a static liquid or hydrostatic pressure. Hydrostatic pressure is most easily addressed by treating the liquid as a continuous distribution of matter, and may be considered a measure of energy per unit volume or energy density. We will further discuss hydrostatic pressure in other sections.

Pressure of an Ideal Gas : This image is a representation of the ideal gas law, as well as the effect of varying the equation parameters on the gas pressure. Pressure within static fluids depends on the properties of the fluid, the acceleration due to gravity, and the depth within the fluid.

Pressure is defined in simplest terms as force per unit area. The derivation of pressure as a measure of energy per unit volume from its definition as force per unit area is given in.

Since, for gases and liquids, the force acting on a system contributing to pressure does not act on a specific point or particular surface, but rather as a distribution of force, analyzing pressure as a measure of energy per unit volume is more appropriate.

For liquids and gases at rest, the pressure of the liquid or gas at any point within the medium is called the hydrostatic pressure. At any such point within a medium, the pressure is the same in all directions, as if the pressure was not the same in all directions, the fluid, whether it is a gas or liquid, would not be static.

Note that the following discussion and expressions pertain only to incompressible fluids at static equilibrium. Energy per Unit Volume : This equation is the derivation of pressure as a measure of energy per unit volume from its definition as force per unit area. The pressure exerted by a static liquid depends only on the depth, density of the liquid, and the acceleration due to gravity. For any given liquid with constant density throughout, pressure increases with increasing depth.

As a result, pressure within a liquid is therefore a function of depth only, with the pressure increasing at a linear rate with respect to increasing depth.

In practical applications involving calculation of pressure as a function of depth, an important distinction must be made as to whether the absolute or relative pressure within a liquid is desired. Equation 2 by itself gives the pressure exerted by a liquid relative to atmospheric pressure, yet if the absolute pressure is desired, the atmospheric pressure must then be added to the pressure exerted by the liquid alone.

When analyzing pressure within gases, a slightly different approach must be taken as, by the nature of gases, the force contributing to pressure arises from the average number of gas molecules occupying a certain point within the gas per unit time. Equation 3 assumes that the gas is incompressible and that the pressure is hydrostatic.

Any region or point, or any static object within a static fluid is in static equilibrium where all forces and torques are equal to zero. Static equilibrium is a particular state of a physical system. It is qualitatively described by an object at rest and by the sum of all forces, with the sum of all torques acting on that object being equal to zero. Static objects are in static equilibrium, with the net force and net torque acting on that object being equal to zero; otherwise there would be a driving mechanism for that object to undergo movement in space.

The analysis and study of objects in static equilibrium and the forces and torques acting on them is called statics—a subtopic of mechanics. Statics is particularly important in the design of static and load bearing structures.

As it pertains to fluidics, static equilibrium concerns the forces acting on a static object within a fluid medium.

For a fluid at rest, the conditions for static equilibrium must be met at any point within the fluid medium. Therefore, the sum of the forces and torques at any point within the static liquid or gas must be zero. Similarly, the sum of the forces and torques of an object at rest within a static fluid medium must also be zero. In considering a stationary object within a liquid medium at rest, the forces acting at any point in time and at any point in space within the medium must be analyzed.

For a stationary object within a static liquid, there are no torques acting on the object so the sum of the torques for such a system is immediately zero; it need not concern analysis since the torque condition for equilibrium is fulfilled. At any point in space within a static fluid, the sum of the acting forces must be zero; otherwise the condition for static equilibrium would not be met.

Next, the forces acting on this region within the medium are taken into account. First, the region has a force of gravity acting downwards its weight equal to its density object, times its volume of the object, times the acceleration due to gravity.

The downward force acting on this region due to the fluid above the region is equal to the pressure times the area of contact. Similarly, there is an upward force acting on this region due to the fluid below the region equal to the pressure times the area of contact. For static equilibrium to be achieved, the sum of these forces must be zero, as shown in. Thus for any region within a fluid, in order to achieve static equilibrium, the pressure from the fluid below the region must be greater than the pressure from the fluid above by the weight of the region.

This force which counteracts the weight of a region or object within a static fluid is called the buoyant force or buoyancy. Static Equilibrium of a Region Within a Fluid : This figure shows the equations for static equilibrium of a region within a fluid.

Region Within a Static Fluid : This figure is a free body diagram of a region within a static fluid. In the case on an object at stationary equilibrium within a static fluid, the sum of the forces acting on that object must be zero. As previously discussed, there are two downward acting forces, one being the weight of the object and the other being the force exerted by the pressure from the fluid above the object. At the same time, there is an upwards force exerted by the pressure from the fluid below the object, which includes the buoyant force.

The appearance of a buoyant force in static fluids is due to the fact that pressure within the fluid changes as depth changes.

The analysis presented above can furthermore be extended to much more complicated systems involving complex objects and diverse materials. By exploiting the fact that pressure is transmitted undiminished in an enclosed static liquid, such as in this type of system, static liquids can be used to transform small amounts of force into large amounts of force for many applications such as hydraulic presses. The cross-sectional area of the bottle changes with height so that at the bottom of the bottle the cross-sectional area is cm 2.

Furthermore, the hydrostatic pressure due to the difference in height of the liquid is given by Equation 1 and yields the total pressure at the bottom surface of the bottle. Since the cross-sectional area at the bottom of the bottle is times larger than at the top, the force contributing to the pressure at the bottom of the bottle is N plus the force from the weight of the static fluid in the bottle. Taking advantage of this phenomenon, hydraulic presses are able to exert a large amount of force requiring a much smaller amount of input force.

Depending on the applied pressure and geometry of the hydraulic press, the magnitude of F 2 can be changed. The difference in height of the fluid between the input and the output ends contributes to the total force exerted by the fluid. For a hydraulic press, the force multiplication factor is the ratio of the output to the input contact areas. Pressure is often measured as gauge pressure, which is defined as the absolute pressure minus the atmospheric pressure. An important distinction must be made as to the type of pressure quantity being used when dealing with pressure measurements and calculations.

Atmospheric pressure is the magnitude of pressure in a system due to the atmosphere, such as the pressure exerted by air molecules a static fluid on the surface of the earth at a given elevation.

In most measurements and calculations, the atmospheric pressure is considered to be constant at 1 atm or , Pa, which is the atmospheric pressure under standard conditions at sea level. Atmospheric pressure is due to the force of the molecules in the atmosphere and is a case of hydrostatic pressure. Depending on the altitude relative to sea level, the actual atmospheric pressure will be less at higher altitudes and more at lower altitudes as the weight of air molecules in the immediate atmosphere changes, thus changing the effective atmospheric pressure.

Atmospheric pressure is a measure of absolute pressure and can be affected by the temperature and air composition of the atmosphere but can generally be accurately approximated to be around standard atmospheric pressure of , Pa.

In this equation p 0 is the pressure at sea level , Pa , g is the acceleration due to gravity, M is the mass of a single molecule of air, R is the universal gas constant, T 0 is the standard temperature at sea level, and h is the height relative to sea level. Pressure and Height : Atmospheric pressure depends on altitude or height. For most applications, particularly those involving pressure measurements, it is more practical to use gauge pressure than absolute pressure as a unit of measurement.

Gauge pressure is a relative pressure measurement which measures pressure relative to atmospheric pressure and is defined as the absolute pressure minus the atmospheric pressure. Most pressure measuring equipment give the pressure of a system in terms of gauge pressure as opposed to absolute pressure.

For example, tire pressure and blood pressure are gauge pressures by convention, while atmospheric pressures, deep vacuum pressures, and altimeter pressures must be absolute. For most working fluids where a fluid exists in a closed system, gauge pressure measurement prevails.

Pressure instruments connected to the system will indicate pressures relative to the current atmospheric pressure. The situation changes when extreme vacuum pressures are measured; absolute pressures are typically used instead. To find the absolute pressure of a system, the atmospheric pressure must then be added to the gauge pressure. While gauge pressure is very useful in practical pressure measurements, most calculations involving pressure, such as the ideal gas law, require pressure values in terms of absolute pressures and thus require gauge pressures to be converted to absolute pressures.

Barometers are devices used for measuring atmospheric and gauge pressure indirectly through the use of hydrostatic fluids. In practice, pressure is most often measured in terms of gauge pressure. Gauge pressure is the pressure of a system above atmospheric pressure. Since atmospheric pressure is mostly constant with little variation near sea level, where most practical pressure measurements are taken, it is assumed to be approximately , Pa. Modern pressure measuring devices sometimes have incorporated mechanisms to account for changes in atmospheric pressure due to elevation changes.

Gauge pressure is much more convenient than absolute pressure for practical measurements and is widely used as an established measure of pressure. However, it is important to determine whether it is necessary to use absolute gauge plus atmospheric pressure for calculations, as is often the case for most calculations, such as those involving the ideal gas law.

Pressure measurements have been accurately taken since the mids with the invention of the traditional barometer. Barometers are devices used to measure pressure and were initially used to measure atmospheric pressure. Early barometers were used to measure atmospheric pressure through the use of hydrostatic fluids. Hydrostatic based barometers consist of columnar devices usually made from glass and filled with a static liquid of consistent density.

The columnar section is sealed, holds a vacuum, and is partially filled with the liquid while the base section is open to the atmosphere and makes an interface with the surrounding environment.

As the atmospheric pressure changes, the pressure exerted by the atmosphere on the fluid reservoir exposed to the atmosphere at the base changes, increasing as the atmospheric pressure increases and decreasing as the atmospheric pressure decreases. This change in pressure causes the height of the fluid in the columnar structure to change, increasing in height as the atmosphere exerts greater pressure on the liquid in the reservoir base and decreasing as the atmosphere exerts lower pressure on the liquid in the reservoir base.

The height of the liquid within the glass column then gives a measure of the atmospheric pressure. Pressure, as determined by hydrostatic barometers, is often measured by determining the height of the liquid in the barometer column, thus the torr as a unit of pressure, but can be used to determine pressure in SI units. Hydrostatic based barometers most commonly use water or mercury as the static liquid. While the use of water is much less hazardous than mercury, mercury is often a better choice for fabricating accurate hydrostatic barometers.

The density of mercury is much higher than that of water, thus allowing for higher accuracy of measurements and the ability to fabricate more compact hydrostatic barometers. In theory, a hydrostatic barometer can be placed in a closed system to measure the absolute pressure and the gauge pressure of the system by subtracting the atmospheric pressure. Another type of barometer is the aneroid barometer, which consists of a small, flexible sealed metal box called an aneroid cell.

The aneroid cell is made from beryllium-copper alloy and is partially evacuated. A stiff spring prevents the aneroid cell from collapsing. Small changes in external air pressure cause the cell to expand or contract.

This expansion and contraction is amplified by mechanical mechanisms to give a pressure reading. Such pressure measuring devices are more practical than hydrostatic barometers for measuring system pressures. Many modern pressure measuring devices are pre-engineered to output gauge pressure measurements. Solids also resist shearing forces. Shearing forces are forces applied tangentially to a surface, as described in Static Equilibrium and Elasticity. Liquids and gases are considered to be fluids because they yield to shearing forces, whereas solids resist them.

Like solids, the molecules in a liquid are bonded to neighboring molecules, but possess many fewer of these bonds. The molecules in a liquid are not locked in place and can move with respect to each other. The distance between molecules is similar to the distances in a solid, and so liquids have definite volumes, but the shape of a liquid changes, depending on the shape of its container.

Gases are not bonded to neighboring atoms and can have large separations between molecules. Gases have neither specific shapes nor definite volumes, since their molecules move to fill the container in which they are held Figure. Figure Forces between the atoms strongly resist attempts to compress the atoms. A gas must be held in a closed container to prevent it from expanding freely and escaping. Liquids deform easily when stressed and do not spring back to their original shape once a force is removed.

This occurs because the atoms or molecules in a liquid are free to slide about and change neighbors. That is, liquids flow so they are a type of fluid , with the molecules held together by mutual attraction. When a liquid is placed in a container with no lid, it remains in the container. Because the atoms are closely packed, liquids, like solids, resist compression; an extremely large force is necessary to change the volume of a liquid. In contrast, atoms in gases are separated by large distances, and the forces between atoms in a gas are therefore very weak, except when the atoms collide with one another.

This makes gases relatively easy to compress and allows them to flow which makes them fluids. When placed in an open container, gases, unlike liquids, will escape. In this chapter, we generally refer to both gases and liquids simply as fluids, making a distinction between them only when they behave differently.

There exists one other phase of matter, plasma , which exists at very high temperatures. At high temperatures, molecules may disassociate into atoms, and atoms disassociate into electrons with negative charges and protons with positive charges , forming a plasma. Plasma will not be discussed in depth in this chapter because plasma has very different properties from the three other common phases of matter, discussed in this chapter, due to the strong electrical forces between the charges.

Suppose a block of brass and a block of wood have exactly the same mass. If both blocks are dropped in a tank of water, why does the wood float and the brass sink Figure? This occurs because the brass has a greater density than water, whereas the wood has a lower density than water. The block of wood is the same in both pictures; it was turned on its side to fit on the scale. Density is an important characteristic of substances. It is crucial, for example, in determining whether an object sinks or floats in a fluid.

The average density of a substance or object is defined as its mass per unit volume,. Figure lists some representative values. The density of gold, for example, is about 2. Density also reveals something about the phase of the matter and its substructure. Notice that the densities of liquids and solids are roughly comparable, consistent with the fact that their atoms are in close contact. The densities of gases are much less than those of liquids and solids, because the atoms in gases are separated by large amounts of empty space.

The gases are displayed for a standard temperature of [latex] 0. The densities of the solids and liquids displayed are given for the standard temperature of [latex] 0. The density of solids and liquids normally increase with decreasing temperature. Figure shows the density of water in various phases and temperature. The density of water increases with decreasing temperature, reaching a maximum at [latex] 4. This behavior of the density of water explains why ice forms at the top of a body of water.

The density of a substance is not necessarily constant throughout the volume of a substance. If the density is constant throughout a substance, the substance is said to be a homogeneous substance.

A solid iron bar is an example of a homogeneous substance. The density is constant throughout, and the density of any sample of the substance is the same as its average density.

If the density of a substance were not constant, the substance is said to be a heterogeneous substance. A chunk of Swiss cheese is an example of a heterogeneous material containing both the solid cheese and gas-filled voids.

Local density at a point is obtained from dividing mass by volume in a small volume around a given point. Local density can be obtained by a limiting process, based on the average density in a small volume around the point in question, taking the limit where the size of the volume approaches zero,.

Since gases are free to expand and contract, the densities of the gases vary considerably with temperature, whereas the densities of liquids vary little with temperature. Therefore, the densities of liquids are often treated as constant, with the density equal to the average density. Density is a dimensional property; therefore, when comparing the densities of two substances, the units must be taken into consideration.

For this reason, a more convenient, dimensionless quantity called the specific gravity is often used to compare densities. Specific gravity is defined as the ratio of the density of the material to the density of water at [latex] 4. Specific gravity, being dimensionless, provides a ready comparison among materials without having to worry about the unit of density. For instance, the density of aluminum is 2.

Specific gravity is a particularly useful quantity with regard to buoyancy, which we will discuss later in this chapter. These are only two of many examples of pressure in fluids. Recall that we introduced the idea of pressure in Static Equilibrium and Elasticity , in the context of bulk stress and strain. Pressure p is defined as the normal force F per unit area A over which the force is applied, or.

A given force can have a significantly different effect, depending on the area over which the force is exerted. Note that although force is a vector, pressure is a scalar. Pressure is a scalar quantity because it is defined to be proportional to the magnitude of the force acting perpendicular to the surface area. The SI unit for pressure is the pascal Pa , named after the French mathematician and physicist Blaise Pascal — , where.

Several other units are used for pressure, which we discuss later in the chapter. Pressure is defined for all states of matter, but it is particularly important when discussing fluids. A force applied perpendicular to the surface compresses or expands the fluid. If you try to compress a fluid, you find that a reaction force develops at each point inside the fluid in the outward direction, balancing the force applied on the molecules at the boundary. Consider a fluid of constant density as shown in Figure.

The pressure due to the fluid is equal to the weight of the fluid divided by the area. The weight of the fluid is equal to its mass times the acceleration due to gravity. The vertical sides cannot exert an upward force on the fluid since it cannot withstand a shearing force , so the bottom must support it all. Since the density is constant, the weight can be calculated using the density:.

The pressure at the bottom of the container is therefore equal to atmospheric pressure added to the weight of the fluid divided by the area:. This equation is only good for pressure at a depth for a fluid of constant density. The pressure at a depth in a fluid of constant density is equal to the pressure of the atmosphere plus the pressure due to the weight of the fluid, or. Consider the pressure and force acting on the dam retaining a reservoir of water Figure. Suppose the dam is m wide and the water is The average pressure p due to the weight of the water is the pressure at the average depth h of Entering the density of water from Figure and taking h to be the average depth of Although this force seems large, it is small compared with the [latex] 1.

In fact, it is only 0. If the reservoir in Figure covered twice the area, but was kept to the same depth, would the dam need to be redesigned? The pressure found in part a of the example is completely independent of the width and length of the lake; it depends only on its average depth at the dam.

In the diagram, note that the thickness of the dam increases with depth to balance the increasing force due to the increasing pressure. A static fluid is a fluid that is not in motion. At any point within a static fluid, the pressure on all sides must be equal—otherwise, the fluid at that point would react to a net force and accelerate. The pressure at any point in a static fluid depends only on the depth at that point. As discussed, pressure in a fluid near Earth varies with depth due to the weight of fluid above a particular level.

In the above examples, we assumed density to be constant and the average density of the fluid to be a good representation of the density. This is a reasonable approximation for liquids like water, where large forces are required to compress the liquid or change the volume.

In a swimming pool, for example, the density is approximately constant, and the water at the bottom is compressed very little by the weight of the water on top. Traveling up in the atmosphere is quite a different situation, however. Fluid located at deeper levels is subjected to more force than fluid nearer to the surface due to the weight of the fluid above it.

Therefore, the pressure calculated at a given depth is different than the pressure calculated using a constant density. Imagine a thin element of fluid at a depth h , as shown in Figure. The weight of the element itself is also shown in the free-body diagram. The weight of the element itself is shown in the free-body diagram. Using a Cartesian y -axis oriented up, we find the following equation for the y -component:. Note that if the element had a non-zero y -component of acceleration, the right-hand side would not be zero but would instead be the mass times the y -acceleration.

The mass of the element can be written in terms of the density of the fluid and the volume of the elements:. This equation tells us that the rate of change of pressure in a fluid is proportional to the density of the fluid. If the range of the depth being analyzed is not too great, we can assume the density to be constant.

But if the range of depth is large enough for the density to vary appreciably, such as in the case of the atmosphere, there is significant change in density with depth. In that case, we cannot use the approximation of a constant density. Note that the pressure in a fluid depends only on the depth from the surface and not on the shape of the container.

Thus, in a container where a fluid can freely move in various parts, the liquid stays at the same level in every part, regardless of the shape, as shown in Figure. In the container pictured, the pressure at the bottom of each column is the same; if it were not the same, the fluid would flow until the pressures became equal.

The change in atmospheric pressure with height is of particular interest. Assuming the temperature of air to be constant, and that the ideal gas law of thermodynamics describes the atmosphere to a good approximation, we can find the variation of atmospheric pressure with height, when the temperature is constant. We discuss the ideal gas law in a later chapter, but we assume you have some familiarity with it from high school and chemistry.

Let p y be the atmospheric pressure at height y. Thus, atmospheric pressure drops exponentially with height, since the y -axis is pointed up from the ground and y has positive values in the atmosphere above sea level. This gives us only a rough estimate of the actual situation, since we have assumed both a constant temperature and a constant g over such great distances from Earth, neither of which is correct in reality. Fluid pressure has no direction, being a scalar quantity, whereas the forces due to pressure have well-defined directions: They are always exerted perpendicular to any surface.

The reason is that fluids cannot withstand or exert shearing forces. Thus, in a static fluid enclosed in a tank, the force exerted on the walls of the tank is exerted perpendicular to the inside surface.

Likewise, pressure is exerted perpendicular to the surfaces of any object within the fluid. Figure illustrates the pressure exerted by air on the walls of a tire and by water on the body of a swimmer.