Density and Pressure _ Boundless Physics.pdf - 8/21/2020...

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Unformatted text preview: 8/21/2020 Density and Pressure | Boundless Physics Boundless Physics Fluids Density and Pressure 1/27 8/21/2020 Density and Pressure | Boundless Physics Alibaba.c Marketp Ad Sourci Suppliers & Ad Alibaba.com Open Pressure Pressure is scalar quantity which is de ned as force per unit area where the force acts in a direction perpendicular to the surface. LEARNING OBJECTIVES Identify factors that determine the pressure exerted by the gas KEY TAKEAWAYS Key Points Pressure is a scalar quantity de ned as force per unit area. Pressure only concerns the force component perpendicular to the surface upon which it acts, thus if the force acts at an angle, the force component along the direction perpendicular to the surface must be used to calculate pressure. 2/27 8/21/2020 Density and Pressure | Boundless Physics The pressure exerted on a surface by an object increases as the weight of the object increases or the surface area of contact decreases. Alternatively the pressure exerted decreases as the weight of the object decreases or the surface area of contact increases. Pressure exerted by ideal gases in con ned containers is due to the average number of collisions of gas molecules with the container walls per unit time. As such, pressure depends on the amount of gas (in number of molecules), its temperature, and the volume of the container. Key Terms ideal gas: Theoretical gas characterized by random motion whose individual molecules do not interact with one another and are chemically inert. kinetic energy: The energy associated with a moving particle or object having a certain mass. Pressure is an important physical quantity—it plays an essential role in topics ranging from thermodynamics to solid and uid mechanics. As a scalar physical quantity (having magnitude but no direction), pressure is de ned as the force per unit area applied perpendicular to the surface to which it is applied. Pressure can be expressed in a number of units depending on the context of use. 3/27 8/21/2020 Density and Pressure | Boundless Physics Representation of Pressure: This image shows the graphical representations and corresponding mathematical expressions for the case in which a force acts perpendicular to the surface of contact, as well as the case in which a force acts at angle θ relative to the surface. Pressure as a Function of Surface Area Since pressure depends only on the force acting perpendicular to the surface upon which it is applied, only the force component perpendicular to the surface contributes to the pressure exerted by that force on that surface. Pressure can be increased by either increasing the force or by decreasing the area or can oppositely be decreased by either decreasing the force or increasing the area. illustrates this concept. A rectangular block weighing 1000 N is rst placed horizontally. It has an area of contact (with the surface upon which it is resting) of 0.1 m2, thus exerting a pressure of 1,000 Pa on that surface. That same block in a di erent con guration (also in Figure 2), in which the block is placed vertically, has an area of contact with the surface upon which it is resting of 0.01 m2, thus exerting a pressure of 10,000 Pa—10 times larger than the rst con guration due to a decrease in the surface area by a factor of 10. Pressure as a Function of Surface Area: Pressure can be increased by either increasing the force or by decreasing 5/27 8/21/2020 Density and Pressure | Boundless Physics the area or can oppositely be decreased by either decreasing the force or increasing the area. A good illustration of this is the reason a sharp knife is far more e ective for cutting than a blunt knife. The same force applied by a sharp knife with a smaller area of contact will exert a much greater pressure than a blunt knife having a considerably larger area of contact. Similarly, a person standing on one leg on a trampoline causes a greater displacement of the trampoline than that same person standing on the same trampoline using two legs—not because the individual exerts a larger force when standing on one leg, but because the area upon which this force is exerted is decreased, thus increasing the pressure on the trampoline. Alternatively, an object having a weight larger than another object of the same dimensionality and area of contact with a given surface will exert a greater pressure on that surface due to an increase in force. Finally, when considering a given force of constant magnitude acting on a constant area of a given surface, the pressure exerted by that force on that surface will be greater the larger the angle of that force as it acts upon the surface, reaching a maximum when that force acts perpendicular to the surface. Liquids and Gases: Fluids Just as a solid exerts a pressure on a surface upon which it is in contact, liquids and gases likewise exert pressures on surfaces and objects upon which they are in contact with. The pressure exerted by an ideal gas on a closed container in which it is con ned is best analyzed on a molecular level. Gas molecules in a gas container move in a random manner throughout the volume of the container, exerting a force on the container walls upon collision. Taking the overall average force of all the collisions of the gas molecules con ned within the container over a unit time allows for a proper measurement of the e ective force of the gas molecules on the container walls. Given that the container acts as a con ning surface for this net force, the gas molecules exert a pressure on the container. For such an ideal gas con ned within a rigid container, the pressure exerted by the gas molecules can be calculated using the ideal gas law: p = nRT V 6/27 8/21/2020 Density and Pressure | Boundless Physics where n is the number of gas molecules, R is the ideal gas constant (R = 8.314 J mol-1 K-1), T is the temperature of the gas, and V is the volume of the container. The pressure exerted by the gas can be increased by: increasing the number of collisions of gas molecules per unit time by increasing the number of gas molecules; increasing the kinetic energy of the gas by increasing the temperature; or decreasing the volume of the container. o ers a representation of the ideal gas law, as well as the e ect of varying the equation parameters on the gas pressure. 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 e ect of varying the equation parameters on the gas pressure. Variation of Pressure With Depth Pressure within static uids depends on the properties of the uid, the acceleration due to gravity, and the depth within the uid. 7/27 8/21/2020 Density and Pressure | Boundless Physics AP Physics 2 - Pressure and … Pressure and Pascal’s Principle: A brief introduction to pressure and Pascal’s Principle, including hydraulics. Units, Equations and Representations In SI units, the unit of pressure is the Pascal (Pa), which is equal to a Newton / meter2 (N/m2). Other important units of pressure include the pound per square inch (psi) and the standard atmosphere (atm). The elementary mathematical expression for pressure is given by: pressure = Force Area = F A where p is pressure, F is the force acting perpendicular to the surface to which this force is applied, and A is the area of the surface. Any object that possesses weight, whether at rest or not, exerts a pressure upon the surface with which it is in contact. The magnitude of the pressure exerted by an object on a given surface is equal to its weight acting in the direction perpendicular to that surface, divided by the total surface area of contact between the object and the surface. shows the graphical representations and corresponding mathematical expressions for the case in which a force acts perpendicular to the surface of contact, as well as the case in which a force acts at angle θ relative to the surface. 4/27 8/21/2020 Density and Pressure | Boundless Physics LEARNING OBJECTIVES Identify factors that determine the pressure exerted by static liquids and gases KEY TAKEAWAYS Key Points Hydrostatic pressure refers to the pressure exerted by a uid (gas or liquid) at any point in space within that uid, assuming that the uid is incompressible and at rest. Pressure within a liquid depends only on the density of the liquid, the acceleration due to gravity, and the depth within the liquid. The pressure exerted by such a static liquid increases linearly with increasing depth. Pressure within a gas depends on the temperature of the gas, the mass of a single molecule of the gas, the acceleration due to gravity, and the height (or depth) within the gas. Key Terms incompressible: Unable to be compressed or condensed. static equilibrium: the physical state in which all components of a system are at rest and the net force is equal to zero throughout the system Pressure is de ned in simplest terms as force per unit area. However, when dealing with pressures exerted by gases and liquids, it is most convenient to approach pressure as a measure of energy per unit volume by means of the de nition of work (W = F·d). The derivation of 8/27 8/21/2020 Density and Pressure | Boundless Physics pressure as a measure of energy per unit volume from its de nition 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 speci c 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 uid, whether it is a gas or liquid, would not be static. Note that the following discussion and expressions pertain only to incompressible uids at static equilibrium. The pressure exerted by a static liquid depends only on the depth, density of the liquid, and the acceleration due to gravity. gives the expression for pressure as a function of depth within an incompressible, static liquid as well as the derivation of this equation from the de nition of pressure as a measure of energy per unit volume (ρ is the density of the gas, g is the acceleration due to gravity, and h is the depth within the Energy per Unit Volume: This equation is the derivation of pressure as a measure of energy per unit volume from its de nition as force per unit area. liquid). For any given liquid with constant density throughout, pressure increases with increasing depth. For example, a person under water at a depth of h1 will experience half the pressure as a person under water at a depth of h2 = 2h1. For many liquids, the density can be assumed to be nearly constant throughout the volume of the liquid and, for virtually all practical applications, so can the acceleration due to gravity (g = 9.81 m/s2). 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. 9/27 8/21/2020 Density and Pressure | Boundless Physics Pressure as Energy per Unit Volume: This equation gives the expression for pressure as a function of depth within an incompressible, static liquid as well as the derivation of this equation from the de nition of pressure as a measure of energy per unit volume (ρ is the density of the gas, g is the acceleration due to gravity, and h is the depth within the liquid). When analyzing pressure within gases, a slightly di erent 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. Thus the force contributing to the pressure of a gas within the medium is not a continuous distribution as for liquids and the barometric equation given in must be utilized to determine the pressure exerted by the gas at a certain depth (or height) within the gas (p0 is the pressure at h = 0, M is the mass of a single molecule of gas, g is the acceleration due to gravity, k is the Boltzmann constant, T is the temperature of the gas, and h is the height or depth within the gas). Equation 3 assumes that the gas is incompressible and that the pressure is hydrostatic. Pressure within a gas: The force contributing to the pressure of a gas within the medium is not a continuous distribution as for liquids and the barometric equation given in this gure must be utilized to determine the pressure exerted by the gas at a certain depth (or height) within the gas (p0 is the pressure at h = 0, M is the mass of a single molecule of gas, g is the acceleration due to gravity, k is the Boltzmann constant, T is the temperature of the gas, and h is the height or depth within the gas) Static Equilibrium 10/27 8/21/2020 Density and Pressure | Boundless Physics Any region or point, or any static object within a static uid is in static equilibrium where all forces and torques are equal to zero. LEARNING OBJECTIVES Identify required conditions for a uid to be in rest KEY TAKEAWAYS Key Points Hydrostatic balance is the term used for a region or stationary object within a static uid which is at static equilibrium, and for which the sum of all forces and sum of all torques is equal to zero. A region or static object within a stationary uid experiences downward forces due to the weight of the region or object, and the pressure exerted from the uid above the region or object, as well as an upward force due to the pressure exerted from the uid below the region or object. For a region or static object within a static uid, the downward force due to the weight of the region or object is counteracted by the upward buoyant force, which is equal to the weight of the uid displaced by the region or object. Key Terms Buoyancy: The power of supporting a body so that it oats; upward pressure exerted by the uid in which a body is immersed. torque: Something that produces or tends to produce torsion or rotation; the moment of a force or system of 11/27 8/21/2020 Density and Pressure | Boundless Physics forces tending to cause rotation. equilibrium: A state of rest or balance due to the equal action of opposing forces. 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 uidics, static equilibrium concerns the forces acting on a static object within a uid medium. Fluids For a uid at rest, the conditions for static equilibrium must be met at any point within the uid 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 uid 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 ful lled. Density At any point in space within a static uid, the sum of the acting forces must be zero; otherwise the condition for static equilibrium would not be met. In analyzing such a simple system, consider a rectangular region within the uid medium with density ρL (same density as the uid medium), width w, length l, and height h, as shown in. Next, the forces acting on this region within the medium are taken into account. First, the 12/27 8/21/2020 Density and Pressure | Boundless Physics 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 uid 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 uid 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 uid, in order to achieve static equilibrium, the pressure from the uid below the region must be greater than the pressure from the uid above by the weight of the region. This force which counteracts the weight of a region or object within a static uid is called the buoyant force (or buoyancy). Static Equilibrium of a Region Within a Fluid: This gure shows the equations for static equilibrium of a region within a uid. Region Within a Static Fluid: This gure is a free body diagram of a region within a static uid. In the case on an object at stationary equilibrium within a static uid, 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 uid above the object. At the same time, there is an upwards force 13/27 8/21/2020 Density and Pressure | Boundless Physics exerted by the pressure from the uid below the object, which includes the buoyant force. shows how the calculation of the forces acting on a stationary object within a static uid would change from those presented in if an object having a density ρS di erent from that of the uid medium is surrounded by the uid. The appearance of a buoyant force in static uids is due to the fact that pressure within the uid changes as depth changes. The analysis presented above can furthermore be extended to much more complicated systems involving complex objects and diverse materials. Pascal’s Principle Pascal’s Principle states that pressure is transmitted and undiminished in a closed static uid. LEARNING OBJECTIVES Apply Pascal’s Principle to describe pressure behavior in static uids KEY TAKEAWAYS Key Points Pascal’s Principle is used to quantitatively relate the pressure at two points in an incompressible, static uid. It states that pressure is transmitted, undiminished, in a closed static uid. The total pressure at any point within an incompressible, static uid is equal to the sum of the applied pressure at any point in that uid and the hydrostatic pressure change due to a di erence in height within that uid. 14/27 8/21/2020 Density and Pressure | Boundless Physics , p2 = p1 + Δp Δp = ρgΔh where p1 is the external applied pressure, ρ is the density of the uid, Δh is the di ere...
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