Misplaced Pages

#244755

88-520: When liquid water becomes steam, it increases in volume by 1,700 times at standard temperature and pressure ; this change in volume can be converted into mechanical work by steam engines such as reciprocating piston type engines and steam turbines , which are a sub-group of steam engines. Piston type steam engines played a central role in the Industrial Revolution and modern steam turbines are used to generate more than 80 % of

176-427: A region D in three-dimensional space is given by the triple or volume integral of the constant function f ( x , y , z ) = 1 {\displaystyle f(x,y,z)=1} over the region. It is usually written as: ∭ D 1 d x d y d z . {\displaystyle \iiint _{D}1\,dx\,dy\,dz.} In cylindrical coordinates ,

264-437: A reservoir , the container's volume is modeled by shapes and calculated using mathematics. To ease calculations, a unit of volume is equal to the volume occupied by a unit cube (with a side length of one). Because the volume occupies three dimensions, if the metre (m) is chosen as a unit of length, the corresponding unit of volume is the cubic metre (m ). The cubic metre is also a SI derived unit . Therefore, volume has

352-457: A unit dimension of L . The metric units of volume uses metric prefixes , strictly in powers of ten . When applying prefixes to units of volume, which are expressed in units of length cubed, the cube operators are applied to the unit of length including the prefix. An example of converting cubic centimetre to cubic metre is: 2.3 cm = 2.3 (cm) = 2.3 (0.01 m) = 0.0000023 m (five zeros). Commonly used prefixes for cubed length units are

440-404: A component opposite to the direction of the displacement at the point of application of the force. For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). If the ball is thrown upwards, the work done by

528-463: A constant force that is not directed along the line, followed by the particle. In this case the dot product F ⋅ d s = F cos θ ds , where θ is the angle between the force vector and the direction of movement, that is W = ∫ C F ⋅ d s = F s cos ⁡ θ . {\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {s} =Fs\cos \theta .} When

616-468: A force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force ), no work is done, since the cosine of 90° is zero. Thus, no work can be performed by gravity on a planet with a circular orbit (this is ideal, as all orbits are slightly elliptical). Also, no work is done on a body moving circularly at a constant speed while constrained by mechanical force, such as moving at constant speed in

704-612: A formula exists for the shape's boundary. Zero- , one- and two-dimensional objects have no volume; in four and higher dimensions, an analogous concept to the normal volume is the hypervolume. The precision of volume measurements in the ancient period usually ranges between 10–50 mL (0.3–2 US fl oz; 0.4–2 imp fl oz). The earliest evidence of volume calculation came from ancient Egypt and Mesopotamia as mathematical problems, approximating volume of simple shapes such as cuboids , cylinders , frustum and cones . These math problems have been written in

792-413: A frictionless ideal centrifuge. Calculating the work as "force times straight path segment" would only apply in the most simple of circumstances, as noted above. If force is changing, or if the body is moving along a curved path, possibly rotating and not necessarily rigid, then only the path of the application point of the force is relevant for the work done, and only the component of the force parallel to

880-486: A measuring unit. The work W done by a constant force of magnitude F on a point that moves a displacement s in a straight line in the direction of the force is the product W = F → ⋅ s → {\displaystyle W={\vec {F}}\cdot {\vec {s}}} For example, if a force of 10 newtons ( F = 10 N ) acts along a point that travels 2 metres ( s = 2 m ), then W = Fs = (10 N) (2 m) = 20 J . This

968-441: A negative value, similar to length and area . Like all continuous monotonic (order-preserving) measures, volumes of bodies can be compared against each other and thus can be ordered. Volume can also be added together and be decomposed indefinitely; the latter property is integral to Cavalieri's principle and to the infinitesimal calculus of three-dimensional bodies. A 'unit' of infinitesimally small volume in integral calculus

SECTION 10

#1732766261245

1056-423: A plentiful supply of steam to spare. Steam engines and steam turbines use the expansion of steam to drive a piston or turbine to perform mechanical work . The ability to return condensed steam as water-liquid to the boiler at high pressure with relatively little expenditure of pumping power is important. Condensation of steam to water often occurs at the low-pressure end of a steam turbine, since this maximizes

1144-452: A potential function is known as potential energy and the forces are said to be conservative . Therefore, work on an object that is merely displaced in a conservative force field , without change in velocity or rotation, is equal to minus the change of potential energy E p of the object, W = − Δ E p . {\displaystyle W=-\Delta E_{\text{p}}.} These formulas show that work

1232-467: A pulley system like the Atwood machine , the internal forces on the rope and at the supporting pulley do no work on the system. Therefore, work need only be computed for the gravitational forces acting on the bodies. Another example is the centripetal force exerted inwards by a string on a ball in uniform circular motion sideways constrains the ball to circular motion restricting its movement away from

1320-454: A supply of steam stored on board in a large tank resembling a conventional locomotive's boiler. This tank was filled by process steam , as is available in many sorts of large factory, such as paper mills . The locomotive's propulsion used pistons and connecting rods, as for a typical steam locomotive. These locomotives were mostly used in places where there was a risk of fire from a boiler's firebox, but were also used in factories that simply had

1408-647: A system at an instant of time. Integration of this power over the trajectory of the point of application, C = x ( t ) , defines the work input to the system by the force. Therefore, the work done by a force F on an object that travels along a curve C is given by the line integral : W = ∫ C F ⋅ d x = ∫ t 1 t 2 F ⋅ v d t , {\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {x} =\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot \mathbf {v} dt,} where dx ( t ) defines

1496-656: Is aligned with the angular velocity vector so that, T = τ S , {\displaystyle \mathbf {T} =\tau \mathbf {S} ,} and both the torque and angular velocity are constant, then the work takes the form, W = ∫ t 1 t 2 τ ϕ ˙ d t = τ ( ϕ 2 − ϕ 1 ) . {\displaystyle W=\int _{t_{1}}^{t_{2}}\tau {\dot {\phi }}\,dt=\tau (\phi _{2}-\phi _{1}).} This result can be understood more simply by considering

1584-416: Is always zero, so the work W = F ⋅ v = 0 , and the magnetic force does not do work. It can change the direction of motion but never change the speed. For moving objects, the quantity of work/time (power) is integrated along the trajectory of the point of application of the force. Thus, at any instant, the rate of the work done by a force (measured in joules/second, or watts ) is the scalar product of

1672-401: Is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. Work is closely related to energy . Energy shares the same unit of measurement with work (Joules) because the energy from the object doing work

1760-738: Is called the potential energy associated with the applied force. The force derived from such a potential function is said to be conservative . Examples of forces that have potential energies are gravity and spring forces. In this case, the gradient of work yields ∇ W = − ∇ U = − ( ∂ U ∂ x , ∂ U ∂ y , ∂ U ∂ z ) = F , {\displaystyle \nabla W=-\nabla U=-\left({\frac {\partial U}{\partial x}},{\frac {\partial U}{\partial y}},{\frac {\partial U}{\partial z}}\right)=\mathbf {F} ,} and

1848-431: Is common for measuring small volume of fluids or granular materials , by using a multiple or fraction of the container. For granular materials, the container is shaken or leveled off to form a roughly flat surface. This method is not the most accurate way to measure volume but is often used to measure cooking ingredients . Air displacement pipette is used in biology and biochemistry to measure volume of fluids at

SECTION 20

#1732766261245

1936-461: Is computed along the trajectory of the rigid body with an angular velocity ω that varies with time, and is therefore said to be path dependent . If the angular velocity vector maintains a constant direction, then it takes the form, ω = ϕ ˙ S , {\displaystyle {\boldsymbol {\omega }}={\dot {\phi }}\mathbf {S} ,} where ϕ {\displaystyle \phi }

2024-425: Is displacement along the line. If F is constant, in addition to being directed along the line, then the integral simplifies further to W = ∫ C F d s = F ∫ C d s = F s {\displaystyle W=\int _{C}F\,ds=F\int _{C}ds=Fs} where s is the displacement of the point along the line. This calculation can be generalized for

2112-466: Is heated further, the droplets evaporate, and at a high enough temperature (which depends on the pressure) all of the water evaporates and the system is in vapour–liquid equilibrium . When steam has reached this equilibrium point, it is referred to as saturated steam . Superheated steam or live steam is steam at a temperature higher than its boiling point for the pressure, which only occurs when all liquid water has evaporated or has been removed from

2200-405: Is necessary to raise body A of 1 pound (libra) to a height of 4 yards (ulnae), as is necessary to raise body B of 4 pounds to a height of 1 yard. In 1759, John Smeaton described a quantity that he called "power" "to signify the exertion of strength, gravitation, impulse, or pressure, as to produce motion." Smeaton continues that this quantity can be calculated if "the weight raised is multiplied by

2288-432: Is the energy transferred to or from an object via the application of force along a displacement . In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. A force is said to do positive work if it has a component in the direction of the displacement of the point of application . A force does negative work if it has

2376-543: Is the joule (J), the same unit as for energy. The ancient Greek understanding of physics was limited to the statics of simple machines (the balance of forces), and did not include dynamics or the concept of work. During the Renaissance the dynamics of the Mechanical Powers , as the simple machines were called, began to be studied from the standpoint of how far they could lift a load, in addition to

2464-634: Is the volume element ; this formulation is useful when working with different coordinate systems , spaces and manifolds . The oldest way to roughly measure a volume of an object is using the human body, such as using hand size and pinches . However, the human body's variations make it extremely unreliable. A better way to measure volume is to use roughly consistent and durable containers found in nature, such as gourds , sheep or pig stomachs , and bladders . Later on, as metallurgy and glass production improved, small volumes nowadays are usually measured using standardized human-made containers. This method

2552-709: Is the angle of rotation about the constant unit vector S . In this case, the work of the torque becomes, W = ∫ t 1 t 2 T ⋅ ω d t = ∫ t 1 t 2 T ⋅ S d ϕ d t d t = ∫ C T ⋅ S d ϕ , {\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot {\boldsymbol {\omega }}\,dt=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot \mathbf {S} {\frac {d\phi }{dt}}dt=\int _{C}\mathbf {T} \cdot \mathbf {S} \,d\phi ,} where C

2640-399: Is the energy associated with the action of a force, so work subsequently possesses the physical dimensions , and units, of energy. The work/energy principles discussed here are identical to electric work/energy principles. Constraint forces determine the object's displacement in the system, limiting it within a range. For example, in the case of a slope plus gravity, the object is stuck to

2728-419: Is the result of a force on a point that follows a curve X , with a velocity v , at each instant. The small amount of work δW that occurs over an instant of time dt is calculated as δ W = F ⋅ d s = F ⋅ v d t {\displaystyle \delta W=\mathbf {F} \cdot d\mathbf {s} =\mathbf {F} \cdot \mathbf {v} dt} where

Steam - Misplaced Pages Continue

2816-440: Is the trajectory from ϕ ( t 1 ) {\displaystyle \phi (t_{1})} to ϕ ( t 2 ) {\displaystyle \phi (t_{2})} . This integral depends on the rotational trajectory ϕ ( t ) {\displaystyle \phi (t)} , and is therefore path-dependent. If the torque τ {\displaystyle \tau }

2904-407: Is the trajectory from x ( t 1 ) to x ( t 2 ). This integral is computed along the trajectory of the particle, and is therefore said to be path dependent . If the force is always directed along this line, and the magnitude of the force is F , then this integral simplifies to W = ∫ C F d s {\displaystyle W=\int _{C}F\,ds} where s

2992-453: Is to Solomon of Caux "that we owe the term work in the sense that it is used in mechanics now". The SI unit of work is the joule (J), named after English physicist James Prescott Joule (1818-1889), which is defined as the work required to exert a force of one newton through a displacement of one metre . The dimensionally equivalent newton-metre (N⋅m) is sometimes used as the measuring unit for work, but this can be confused with

3080-693: Is tradition to define this function with a negative sign so that positive work is a reduction in the potential, that is W = ∫ C F ⋅ d x = ∫ x ( t 1 ) x ( t 2 ) F ⋅ d x = U ( x ( t 1 ) ) − U ( x ( t 2 ) ) . {\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {x} =\int _{\mathbf {x} (t_{1})}^{\mathbf {x} (t_{2})}\mathbf {F} \cdot d\mathbf {x} =U(\mathbf {x} (t_{1}))-U(\mathbf {x} (t_{2})).} The function U ( x )

3168-401: Is transferred to the other objects it interacts with when work is being done. The work–energy principle states that an increase in the kinetic energy of a rigid body is caused by an equal amount of positive work done on the body by the resultant force acting on that body. Conversely, a decrease in kinetic energy is caused by an equal amount of negative work done by the resultant force. Thus, if

3256-428: Is used for soil sterilization to avoid the use of harmful chemical agents and increase soil health . Steam's capacity to transfer heat is also used in the home: for cooking vegetables, steam cleaning of fabric, carpets and flooring, and for heating buildings. In each case, water is heated in a boiler, and the steam carries the energy to a target object. Steam is also used in ironing clothes to add enough humidity with

3344-494: Is used in various chemical processes as a reactant. Steam cracking of long chain hydrocarbons produces lower molecular weight hydrocarbons for fuel or other chemical applications. Steam reforming produces syngas or hydrogen . Used in cleaning of fibers and other materials, sometimes in preparation for painting. Steam is also useful in melting hardened grease and oil residues, so it is useful in cleaning kitchen floors and equipment and internal combustion engines and parts. Among

3432-503: Is used when integrating by an axis parallel to the axis of rotation. The general equation can be written as: V = π ∫ a b | f ( x ) 2 − g ( x ) 2 | d x {\displaystyle V=\pi \int _{a}^{b}\left|f(x)^{2}-g(x)^{2}\right|\,dx} where f ( x ) {\textstyle f(x)} and g ( x ) {\textstyle g(x)} are

3520-692: The F ⋅ v is the power over the instant dt . The sum of these small amounts of work over the trajectory of the point yields the work, W = ∫ t 1 t 2 F ⋅ v d t = ∫ t 1 t 2 F ⋅ d s d t d t = ∫ C F ⋅ d s , {\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot \mathbf {v} \,dt=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot {\tfrac {d\mathbf {s} }{dt}}\,dt=\int _{C}\mathbf {F} \cdot d\mathbf {s} ,} where C

3608-399: The T ⋅ ω is the power over the instant dt . The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = ∫ t 1 t 2 T ⋅ ω d t . {\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot {\boldsymbol {\omega }}\,dt.} This integral

Steam - Misplaced Pages Continue

3696-729: The Moscow Mathematical Papyrus (c. 1820 BCE). In the Reisner Papyrus , ancient Egyptians have written concrete units of volume for grain and liquids, as well as a table of length, width, depth, and volume for blocks of material. The Egyptians use their units of length (the cubit , palm , digit ) to devise their units of volume, such as the volume cubit or deny (1 cubit × 1 cubit × 1 cubit), volume palm (1 cubit × 1 cubit × 1 palm), and volume digit (1 cubit × 1 cubit × 1 digit). The last three books of Euclid's Elements , written in around 300 BCE, detailed

3784-410: The cube , cuboid and cylinder , they have an essentially the same volume calculation formula as one for the prism : the base of the shape multiplied by its height . The calculation of volume is a vital part of integral calculus. One of which is calculating the volume of solids of revolution , by rotating a plane curve around a line on the same plane. The washer or disc integration method

3872-443: The cubic metre and litre ) or by various imperial or US customary units (such as the gallon , quart , cubic inch ). The definition of length and height (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. By metonymy ,

3960-575: The energy efficiency , but such wet-steam conditions must be limited to avoid excessive turbine blade erosion. Engineers use an idealised thermodynamic cycle , the Rankine cycle , to model the behaviour of steam engines. Steam turbines are often used in the production of electricity. An autoclave , which uses steam under pressure, is used in microbiology laboratories and similar environments for sterilization . Steam, especially dry (highly superheated) steam, may be used for antimicrobial cleaning even to

4048-593: The imperial gallon was defined to be the volume occupied by ten pounds of water at 17 °C (62 °F). This definition was further refined until the United Kingdom's Weights and Measures Act 1985 , which makes 1 imperial gallon precisely equal to 4.54609 litres with no use of water. The 1960 redefinition of the metre from the International Prototype Metre to the orange-red emission line of krypton-86 atoms unbounded

4136-522: The sester , amber , coomb , and seam . The sheer quantity of such units motivated British kings to standardize them, culminated in the Assize of Bread and Ale statute in 1258 by Henry III of England . The statute standardized weight, length and volume as well as introduced the peny, ounce, pound, gallon and bushel. In 1618, the London Pharmacopoeia (medicine compound catalog) adopted

4224-439: The volume integral is ∭ D r d r d θ d z , {\displaystyle \iiint _{D}r\,dr\,d\theta \,dz,} In spherical coordinates (using the convention for angles with θ {\displaystyle \theta } as the azimuth and φ {\displaystyle \varphi } measured from the polar axis; see more on conventions ),

4312-491: The Roman gallon or congius as a basic unit of volume and gave a conversion table to the apothecaries' units of weight. Around this time, volume measurements are becoming more precise and the uncertainty is narrowed to between 1–5 mL (0.03–0.2 US fl oz; 0.04–0.2 imp fl oz). Around the early 17th century, Bonaventura Cavalieri applied the philosophy of modern integral calculus to calculate

4400-401: The advantages of using steam versus a hot water spray are the facts that steam can operate at higher temperatures and it uses substantially less water per minute. [REDACTED] Wikiversity has steam tables with figures and Matlab code Volume Volume is a measure of regions in three-dimensional space . It is often quantified numerically using SI derived units (such as

4488-409: The application point velocity is doing work (positive work when in the same direction, and negative when in the opposite direction of the velocity). This component of force can be described by the scalar quantity called scalar tangential component ( F cos( θ ) , where θ is the angle between the force and the velocity). And then the most general definition of work can be formulated as follows: If

SECTION 50

#1732766261245

4576-429: The centre of the circle. This force does zero work because it is perpendicular to the velocity of the ball. The magnetic force on a charged particle is F = q v × B , where q is the charge, v is the velocity of the particle, and B is the magnetic field . The result of a cross product is always perpendicular to both of the original vectors, so F ⊥ v . The dot product of two perpendicular vectors

4664-474: The constraint. Thus the virtual work done by the forces of constraint is zero, a result which is only true if friction forces are excluded. Fixed, frictionless constraint forces do not perform work on the system, as the angle between the motion and the constraint forces is always 90° . Examples of workless constraints are: rigid interconnections between particles, sliding motion on a frictionless surface, and rolling contact without slipping. For example, in

4752-487: The contained volume does not need to fill towards the container's capacity, or vice versa. Containers can only hold a specific amount of physical volume, not weight (excluding practical concerns). For example, a 50,000 bbl (7,900,000 L) tank that can just hold 7,200 t (15,900,000 lb) of fuel oil will not be able to contain the same 7,200 t (15,900,000 lb) of naphtha , due to naphtha's lower density and thus larger volume. For many shapes such as

4840-399: The cubic millimetre (mm ), cubic centimetre (cm ), cubic decimetre (dm ), cubic metre (m ) and the cubic kilometre (km ). The conversion between the prefix units are as follows: 1000 mm = 1 cm , 1000 cm = 1 dm , and 1000 dm = 1 m . The metric system also includes the litre (L) as a unit of volume, where 1 L = 1 dm = 1000 cm = 0.001 m . For

4928-638: The electric generation cycle. The world's biggest steam generation system is the New York City steam system , which pumps steam into 100,000 buildings in Manhattan from seven co-generation plants. In other industrial applications steam is used for energy storage , which is introduced and extracted by heat transfer, usually through pipes. Steam is a capacious reservoir for thermal energy because of water's high heat of vaporization . Fireless steam locomotives were steam locomotives that operated from

5016-533: The exact formulas for calculating the volume of parallelepipeds , cones, pyramids , cylinders, and spheres . The formula were determined by prior mathematicians by using a primitive form of integration , by breaking the shapes into smaller and simpler pieces. A century later, Archimedes ( c.  287 – 212 BCE ) devised approximate volume formula of several shapes using the method of exhaustion approach, meaning to derive solutions from previous known formulas from similar shapes. Primitive integration of shapes

5104-416: The force (a vector), and the velocity vector of the point of application. This scalar product of force and velocity is known as instantaneous power . Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus , the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application. Work

5192-549: The force is variable, then work is given by the line integral : W = ∫ F → ⋅ d s → {\displaystyle W=\int {\vec {F}}\cdot d{\vec {s}}} where d s → {\displaystyle d{\vec {s}}} is the tiny change in displacement vector. Work is a scalar quantity , so it has only magnitude and no direction. Work transfers energy from one place to another, or one form to another. The SI unit of work

5280-498: The force they could apply, leading eventually to the new concept of mechanical work. The complete dynamic theory of simple machines was worked out by Italian scientist Galileo Galilei in 1600 in Le Meccaniche ( On Mechanics ), in which he showed the underlying mathematical similarity of the machines as force amplifiers. He was the first to explain that simple machines do not create energy, only transform it. Although work

5368-434: The force varies (e.g. compressing a spring) we need to use calculus to find the work done. If the force as a variable of x is given by F ( x ) , then the work done by the force along the x-axis from x 1 to x 2 is: Thus, the work done for a variable force can be expressed as a definite integral of force over displacement. If the displacement as a variable of time is given by ∆ x (t) , then work done by

SECTION 60

#1732766261245

5456-570: The golden crown to find its volume, and thus its density and purity, due to the extreme precision involved. Instead, he likely have devised a primitive form of a hydrostatic balance . Here, the crown and a chunk of pure gold with a similar weight are put on both ends of a weighing scale submerged underwater, which will tip accordingly due to the Archimedes' principle . In the Middle Ages , many units for measuring volume were made, such as

5544-525: The gravitational force is negative, and is equal to the weight multiplied by the displacement in the upwards direction. Both force and displacement are vectors . The work done is given by the dot product of the two vectors, where the result is a scalar . When the force F is constant and the angle θ between the force and the displacement s is also constant, then the work done is given by: W = F s cos ⁡ θ {\displaystyle W=Fs\cos {\theta }} If

5632-464: The heat to take wrinkles out and put intentional creases into the clothing. As of 2000 around 90% of all electricity was generated using steam as the working fluid , nearly all by steam turbines. In electric generation, steam is typically condensed at the end of its expansion cycle, and returned to the boiler for re-use. However, in co-generation , steam is piped into buildings through a district heating system to provide heat energy after its use in

5720-521: The height to which it can be raised in a given time," making this definition remarkably similar to Coriolis 's. According to the 1957 physics textbook by Max Jammer , the term work was introduced in 1826 by the French mathematician Gaspard-Gustave Coriolis as "weight lifted through a height", which is based on the use of early steam engines to lift buckets of water out of flooded ore mines. According to Rene Dugas, French engineer and historian, it

5808-407: The levels of sterilization. Steam is a non-toxic antimicrobial agent. Steam is used in piping for utility lines. It is also used in jacketing and tracing of piping to maintain the uniform temperature in pipelines and vessels. Steam is used across multiple industries for its ability to transfer heat to drive chemical reactions, sterilize or disinfect objects and to maintain constant temperatures. In

5896-415: The litre unit, the commonly used prefixes are the millilitre (mL), centilitre (cL), and the litre (L), with 1000 mL = 1 L, 10 mL = 1 cL, 10 cL = 1 dL, and 10 dL = 1 L. Various other imperial or U.S. customary units of volume are also in use, including: Capacity is the maximum amount of material that a container can hold, measured in volume or weight . However,

5984-410: The lumber industry, steam is used in the process of wood bending , killing insects, and increasing plasticity. Steam is used to accentuate drying of concrete especially in prefabricates. Care should be taken since concrete produces heat during hydration and additional heat from the steam could be detrimental to hardening reaction processes of the concrete. In chemical and petrochemical industries , steam

6072-510: The measurement unit of torque . Usage of N⋅m is discouraged by the SI authority , since it can lead to confusion as to whether the quantity expressed in newton-metres is a torque measurement, or a measurement of work. Another unit for work is the foot-pound , which comes from the English system of measurement. As the unit name suggests, it is the product of pounds for the unit of force and feet for

6160-597: The metre, cubic metre, and litre from physical objects. This also make the metre and metre-derived units of volume resilient to changes to the International Prototype Metre. The definition of the metre was redefined again in 1983 to use the speed of light and second (which is derived from the caesium standard ) and reworded for clarity in 2019 . As a measure of the Euclidean three-dimensional space , volume cannot be physically measured as

6248-598: The microscopic scale. Calibrated measuring cups and spoons are adequate for cooking and daily life applications, however, they are not precise enough for laboratories . There, volume of liquids is measured using graduated cylinders , pipettes and volumetric flasks . The largest of such calibrated containers are petroleum storage tanks , some can hold up to 1,000,000  bbl (160,000,000 L) of fluids. Even at this scale, by knowing petroleum's density and temperature, very precise volume measurement in these tanks can still be made. For even larger volumes such as in

6336-456: The modern integral calculus, which remains in use in the 21st century. On 7 April 1795, the metric system was formally defined in French law using six units. Three of these are related to volume: the stère  (1 m ) for volume of firewood; the litre  (1 dm ) for volumes of liquid; and the gramme , for mass—defined as the mass of one cubic centimetre of water at the temperature of melting ice. Thirty years later in 1824,

6424-589: The net work is positive, then the particle's kinetic energy increases by the amount of the work. If the net work done is negative, then the particle's kinetic energy decreases by the amount of work. From Newton's second law , it can be shown that work on a free (no fields), rigid (no internal degrees of freedom) body, is equal to the change in kinetic energy E k corresponding to the linear velocity and angular velocity of that body, W = Δ E k . {\displaystyle W=\Delta E_{\text{k}}.} The work of forces generated by

6512-409: The plane curve boundaries. The shell integration method is used when integrating by an axis perpendicular to the axis of rotation. The equation can be written as: V = 2 π ∫ a b x | f ( x ) − g ( x ) | d x {\displaystyle V=2\pi \int _{a}^{b}x|f(x)-g(x)|\,dx} The volume of

6600-408: The slope and, when attached to a taut string, it cannot move in an outwards direction to make the string any 'tauter'. It eliminates all displacements in that direction, that is, the velocity in the direction of the constraint is limited to 0, so that the constraint forces do not perform work on the system. For a mechanical system , constraint forces eliminate movement in directions that characterize

6688-470: The system. Steam tables contain thermodynamic data for water/saturated steam and are often used by engineers and scientists in design and operation of equipment where thermodynamic cycles involving steam are used. Additionally, thermodynamic phase diagrams for water/steam, such as a temperature-entropy diagram or a Mollier diagram shown in this article, may be useful. Steam charts are also used for analysing thermodynamic cycles. In agriculture , steam

6776-418: The term "volume" sometimes is used to refer to the corresponding region (e.g., bounding volume ). In ancient times, volume was measured using similar-shaped natural containers. Later on, standardized containers were used. Some simple three-dimensional shapes can have their volume easily calculated using arithmetic formulas . Volumes of more complicated shapes can be calculated with integral calculus if

6864-417: The torque τ = Fr , to obtain W = F r ϕ = τ ϕ , {\displaystyle W=Fr\phi =\tau \phi ,} as presented above. Notice that only the component of torque in the direction of the angular velocity vector contributes to the work. The scalar product of a force F and the velocity v of its point of application defines the power input to

6952-482: The torque as arising from a force of constant magnitude F , being applied perpendicularly to a lever arm at a distance r {\displaystyle r} , as shown in the figure. This force will act through the distance along the circular arc l = s = r ϕ {\displaystyle l=s=r\phi } , so the work done is W = F s = F r ϕ . {\displaystyle W=Fs=Fr\phi .} Introduce

7040-488: The trajectory C and v is the velocity along this trajectory. In general this integral requires that the path along which the velocity is defined, so the evaluation of work is said to be path dependent. The time derivative of the integral for work yields the instantaneous power, d W d t = P ( t ) = F ⋅ v . {\displaystyle {\frac {dW}{dt}}=P(t)=\mathbf {F} \cdot \mathbf {v} .} If

7128-427: The unit of displacement. One joule is equivalent to 0.07376 ft-lbs. Non-SI units of work include the newton-metre, erg , the foot-pound, the foot-poundal , the kilowatt hour , the litre-atmosphere , and the horsepower-hour . Due to work having the same physical dimension as heat , occasionally measurement units typically reserved for heat or energy content, such as therm , BTU and calorie , are used as

7216-587: The variable force from t 1 to t 2 is: Thus, the work done for a variable force can be expressed as a definite integral of power over time. A force couple results from equal and opposite forces, acting on two different points of a rigid body. The sum (resultant) of these forces may cancel, but their effect on the body is the couple or torque T . The work of the torque is calculated as δ W = T ⋅ ω d t , {\displaystyle \delta W=\mathbf {T} \cdot {\boldsymbol {\omega }}\,dt,} where

7304-475: The volume integral is ∭ D ρ 2 sin ⁡ φ d ρ d θ d φ . {\displaystyle \iiint _{D}\rho ^{2}\sin \varphi \,d\rho \,d\theta \,d\varphi .} A polygon mesh is a representation of the object's surface, using polygons . The volume mesh explicitly define its volume and surface properties. Mechanical work In science, work

7392-399: The volume of any object. He devised Cavalieri's principle , which said that using thinner and thinner slices of the shape would make the resulting volume more and more accurate. This idea would then be later expanded by Pierre de Fermat , John Wallis , Isaac Barrow , James Gregory , Isaac Newton , Gottfried Wilhelm Leibniz and Maria Gaetana Agnesi in the 17th and 18th centuries to form

7480-435: The work for an applied force is independent of the path, then the work done by the force, by the gradient theorem , defines a potential function which is evaluated at the start and end of the trajectory of the point of application. This means that there is a potential function U ( x ) , that can be evaluated at the two points x ( t 1 ) and x ( t 2 ) to obtain the work over any trajectory between these two points. It

7568-400: The world's electricity. If liquid water comes in contact with a very hot surface or depressurizes quickly below its vapour pressure , it can create a steam explosion . Steam is traditionally created by heating a boiler via burning coal and other fuels, but it is also possible to create steam with solar energy. Water vapour that includes water droplets is described as wet steam . As wet steam

7656-501: Was also discovered independently by Liu Hui in the 3rd century CE, Zu Chongzhi in the 5th century CE, the Middle East and India . Archimedes also devised a way to calculate the volume of an irregular object, by submerging it underwater and measure the difference between the initial and final water volume. The water volume difference is the volume of the object. Though highly popularized, Archimedes probably does not submerge

7744-533: Was not formally used until 1826, similar concepts existed before then. Early names for the same concept included moment of activity, quantity of action, latent live force, dynamic effect, efficiency , and even force . In 1637, the French philosopher René Descartes wrote: Lifting 100 lb one foot twice over is the same as lifting 200 lb one foot, or 100 lb two feet. In 1686, the German philosopher Gottfried Leibniz wrote: The same force ["work" in modern terms]

#244755