In physics, energy is the quantitative property that must be transferred to an object in order to perform work on, or to heat, the object. Energy is a conserved quantity; the SI unit of energy is the joule, the energy transferred to an object by the work of moving it a distance of 1 metre against a force of 1 newton. Common forms of energy include the kinetic energy of a moving object, the potential energy stored by an object's position in a force field, the elastic energy stored by stretching solid objects, the chemical energy released when a fuel burns, the radiant energy carried by light, the thermal energy due to an object's temperature. Mass and energy are related. Due to mass–energy equivalence, any object that has mass when stationary has an equivalent amount of energy whose form is called rest energy, any additional energy acquired by the object above that rest energy will increase the object's total mass just as it increases its total energy. For example, after heating an object, its increase in energy could be measured as a small increase in mass, with a sensitive enough scale.
Living organisms require exergy to stay alive, such as the energy. Human civilization requires energy to function, which it gets from energy resources such as fossil fuels, nuclear fuel, or renewable energy; the processes of Earth's climate and ecosystem are driven by the radiant energy Earth receives from the sun and the geothermal energy contained within the earth. The total energy of a system can be subdivided and classified into potential energy, kinetic energy, or combinations of the two in various ways. Kinetic energy is determined by the movement of an object – or the composite motion of the components of an object – and potential energy reflects the potential of an object to have motion, is a function of the position of an object within a field or may be stored in the field itself. While these two categories are sufficient to describe all forms of energy, it is convenient to refer to particular combinations of potential and kinetic energy as its own form. For example, macroscopic mechanical energy is the sum of translational and rotational kinetic and potential energy in a system neglects the kinetic energy due to temperature, nuclear energy which combines utilize potentials from the nuclear force and the weak force), among others.
The word energy derives from the Ancient Greek: translit. Energeia, lit.'activity, operation', which appears for the first time in the work of Aristotle in the 4th century BC. In contrast to the modern definition, energeia was a qualitative philosophical concept, broad enough to include ideas such as happiness and pleasure. In the late 17th century, Gottfried Leibniz proposed the idea of the Latin: vis viva, or living force, which defined as the product of the mass of an object and its velocity squared. To account for slowing due to friction, Leibniz theorized that thermal energy consisted of the random motion of the constituent parts of matter, although it would be more than a century until this was accepted; the modern analog of this property, kinetic energy, differs from vis viva only by a factor of two. In 1807, Thomas Young was the first to use the term "energy" instead of vis viva, in its modern sense. Gustave-Gaspard Coriolis described "kinetic energy" in 1829 in its modern sense, in 1853, William Rankine coined the term "potential energy".
The law of conservation of energy was first postulated in the early 19th century, applies to any isolated system. It was argued for some years whether heat was a physical substance, dubbed the caloric, or a physical quantity, such as momentum. In 1845 James Prescott Joule discovered the generation of heat; these developments led to the theory of conservation of energy, formalized by William Thomson as the field of thermodynamics. Thermodynamics aided the rapid development of explanations of chemical processes by Rudolf Clausius, Josiah Willard Gibbs, Walther Nernst, it led to a mathematical formulation of the concept of entropy by Clausius and to the introduction of laws of radiant energy by Jožef Stefan. According to Noether's theorem, the conservation of energy is a consequence of the fact that the laws of physics do not change over time. Thus, since 1918, theorists have understood that the law of conservation of energy is the direct mathematical consequence of the translational symmetry of the quantity conjugate to energy, namely time.
In 1843, James Prescott Joule independently discovered the mechanical equivalent in a series of experiments. The most famous of them used the "Joule apparatus": a descending weight, attached to a string, caused rotation of a paddle immersed in water insulated from heat transfer, it showed that the gravitational potential energy lost by the weight in descending was equal to the internal energy gained by the water through friction with the paddle. In the International System of Units, the unit of energy is the joule, named after James Prescott Joule, it is a derived unit. It is equal to the energy expended in applying a force of one newton through a distance of one metre; however energy is expressed in many other units not part of the SI, such as ergs, British Thermal Units, kilowatt-hours and kilocalories, which require a conversion factor when expressed in SI units. The SI unit of energy rate is the watt, a joule per second. Thus, one joule is one watt-second, 3600 joules equal one wa
In physics, a force is said to do work if, when acting, there is a displacement of the point of application in the direction of the force. For example, when a ball is held above the ground and dropped, the work done on the ball as it falls is equal to the weight of the ball multiplied by the distance to the ground; when the force is constant and the angle between the force and the displacement is θ the work done is given by W = Fs cos θ. Work transfers energy from one form to another. According to Jammer, the term work was introduced in 1826 by the French mathematician Gaspard-Gustave Coriolis as "weight lifted through a height", 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 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. The SI unit of work is the joule, defined as the work expended by a force of one newton through a displacement of one metre.
The dimensionally equivalent newton-metre is sometimes used as the measuring unit for work, but this can be confused with the unit newton-metre, 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. Non-SI units of work include the newton-metre, the foot-pound, the foot-poundal, the kilowatt hour, the litre-atmosphere, the horsepower-hour. Due to work having the same physical dimension as heat measurement units reserved for heat or energy content, such as therm, BTU and Calorie, are utilized as 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. For example, if a force of 10 newtons acts along a point that travels 2 metres W = F s = = 20 J; this is 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 related to energy; 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. From Newton's second law, it can be shown that work on a free, rigid body, is equal to the change in kinetic energy K E of the linear velocity and angular velocity of that body, W = Δ K E; the work of forces generated by a potential function is known as potential energy and the forces are said to be conservative. Therefore, work on an object, displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy P E of the object, W = − Δ P E; these formulas show that work is the energy associated with the action of a force, so work subsequently possesses the physical dimensions, units, of energy.
The work/energy principles discussed here are identical to Electric work/energy principles. Constraint forces limit the movement of components in a system, such as constraining an object to a surface. Constraint forces restrict the velocity in the direction of the constraint to zero, which means the constraint forces do not perform work on the system. For a mechanical system, constraint forces eliminate movement in directions that characterize the constraint, thus constraint forces do not perform work on the system, because the component of velocity along the constraint force at each point of application is zero. For example, in 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 gravity forces acting on the bodies. For example, 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 the center of the circle.
This force does zero work. Another example is a book on a table. If external forces are applied to the book so that it slides on the table the force exerted by the table constrains the book from moving downwards; the force exerted by the table supports the book and is perpendicular to its movement which means that this constraint force does not perform work. The magnetic force on a charged particle is F = qv × B, where q is the charge, v is the velocity of the particle, B is the mag
Rigid body dynamics
Rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are rigid, which means that they do not deform under the action of applied forces, simplifies the analysis by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body; this excludes bodies that display fluid elastic, plastic behavior. The dynamics of a rigid body system is described by the laws of kinematics and by the application of Newton's second law or their derivative form Lagrangian mechanics; the solution of these equations of motion provides a description of the position, the motion and the acceleration of the individual components of the system and overall the system itself, as a function of time. The formulation and solution of rigid body dynamics is an important tool in the computer simulation of mechanical systems. If a system of particles moves parallel to a fixed plane, the system is said to be constrained to planar movement.
In this case, Newton's laws for a rigid system of N particles, Pi, i=1... N, simplify. Determine the resultant force and torque at a reference point R, to obtain F = ∑ i = 1 N m i A i, T = ∑ i = 1 N ×, where ri denotes the planar trajectory of each particle; the kinematics of a rigid body yields the formula for the acceleration of the particle Pi in terms of the position R and acceleration A of the reference particle as well as the angular velocity vector ω and angular acceleration vector α of the rigid system of particles as, A i = α × + ω × + A. For systems that are constrained to planar movement, the angular velocity and angular acceleration vectors are directed along k perpendicular to the plane of movement, which simplifies this acceleration equation. In this case, the acceleration vectors can be simplified by introducing the unit vectors ei from the reference point R to a point ri and the unit vectors t i = k × e i, so A i = α − ω 2 + A; this yields the resultant force on the system as F = α ∑ i = 1 N m i − ω 2 ∑ i = 1 N m i + A, torque as T = ∑ i = 1 N × = α k → + × A, where e i × e i = 0 and e i × t i = k is the unit
Time is the indefinite continued progress of existence and events that occur in irreversible succession through the past, in the present, the future. Time is a component quantity of various measurements used to sequence events, to compare the duration of events or the intervals between them, to quantify rates of change of quantities in material reality or in the conscious experience. Time is referred to as a fourth dimension, along with three spatial dimensions. Time has long been an important subject of study in religion and science, but defining it in a manner applicable to all fields without circularity has eluded scholars. Diverse fields such as business, sports, the sciences, the performing arts all incorporate some notion of time into their respective measuring systems. Time in physics is unambiguously operationally defined as "what a clock reads". See Units of Time. Time is one of the seven fundamental physical quantities in both the International System of Units and International System of Quantities.
Time is used to define other quantities – such as velocity – so defining time in terms of such quantities would result in circularity of definition. An operational definition of time, wherein one says that observing a certain number of repetitions of one or another standard cyclical event constitutes one standard unit such as the second, is useful in the conduct of both advanced experiments and everyday affairs of life; the operational definition leaves aside the question whether there is something called time, apart from the counting activity just mentioned, that flows and that can be measured. Investigations of a single continuum called spacetime bring questions about space into questions about time, questions that have their roots in the works of early students of natural philosophy. Temporal measurement has occupied scientists and technologists, was a prime motivation in navigation and astronomy. Periodic events and periodic motion have long served as standards for units of time. Examples include the apparent motion of the sun across the sky, the phases of the moon, the swing of a pendulum, the beat of a heart.
The international unit of time, the second, is defined by measuring the electronic transition frequency of caesium atoms. Time is of significant social importance, having economic value as well as personal value, due to an awareness of the limited time in each day and in human life spans. Speaking, methods of temporal measurement, or chronometry, take two distinct forms: the calendar, a mathematical tool for organising intervals of time, the clock, a physical mechanism that counts the passage of time. In day-to-day life, the clock is consulted for periods less than a day whereas the calendar is consulted for periods longer than a day. Personal electronic devices display both calendars and clocks simultaneously; the number that marks the occurrence of a specified event as to hour or date is obtained by counting from a fiducial epoch – a central reference point. Artifacts from the Paleolithic suggest that the moon was used to reckon time as early as 6,000 years ago. Lunar calendars were among the first to appear, with years of either 13 lunar months.
Without intercalation to add days or months to some years, seasons drift in a calendar based on twelve lunar months. Lunisolar calendars have a thirteenth month added to some years to make up for the difference between a full year and a year of just twelve lunar months; the numbers twelve and thirteen came to feature prominently in many cultures, at least due to this relationship of months to years. Other early forms of calendars originated in Mesoamerica in ancient Mayan civilization; these calendars were religiously and astronomically based, with 18 months in a year and 20 days in a month, plus five epagomenal days at the end of the year. The reforms of Julius Caesar in 45 BC put the Roman world on a solar calendar; this Julian calendar was faulty in that its intercalation still allowed the astronomical solstices and equinoxes to advance against it by about 11 minutes per year. Pope Gregory XIII introduced a correction in 1582. During the French Revolution, a new clock and calendar were invented in attempt to de-Christianize time and create a more rational system in order to replace the Gregorian calendar.
The French Republican Calendar's days consisted of ten hours of a hundred minutes of a hundred seconds, which marked a deviation from the 12-based duodecimal system used in many other devices by many cultures. The system was abolished in 1806. A large variety of devices have been invented to measure time; the study of these devices is called horology. An Egyptian device that dates to c. 1500 BC, similar in shape to a bent T-square, measured the passage of time from the shadow cast by its crossbar on a nonlinear rule. The T was oriented eastward in the mornings. At noon, the device was turned around so. A sundial uses a gnomon to cast a shadow on a set of markings calibrated to the hour; the position of the shadow marks the hour in local time. The idea to separate the day into smaller parts is credited to Egyptians because of their sundials, which operated on a duodecimal system; the importance of the number 12 is due to the number of lunar cycles in a year and the number of stars used to count the passage of night.
The most precise timekeeping device of the ancient
In physics, power is the rate of doing work or of transferring heat, i.e. the amount of energy transferred or converted per unit time. Having no direction, it is a scalar quantity. In the International System of Units, the unit of power is the joule per second, known as the watt in honour of James Watt, the eighteenth-century developer of the condenser steam engine. Another common and traditional measure is horsepower. Being the rate of work, the equation for power can be written: power = work time As a physical concept, power requires both a change in the physical system and a specified time in which the change occurs; this is distinct from the concept of work, only measured in terms of a net change in the state of the physical system. The same amount of work is done when carrying a load up a flight of stairs whether the person carrying it walks or runs, but more power is needed for running because the work is done in a shorter amount of time; the output power of an electric motor is the product of the torque that the motor generates and the angular velocity of its output shaft.
The power involved in moving a vehicle is the product of the traction force of the wheels and the velocity of the vehicle. The rate at which a light bulb converts electrical energy into light and heat is measured in watts—the higher the wattage, the more power, or equivalently the more electrical energy is used per unit time; the dimension of power is energy divided by time. The SI unit of power is the watt, equal to one joule per second. Other units of power include ergs per second, metric horsepower, foot-pounds per minute. One horsepower is equivalent to 33,000 foot-pounds per minute, or the power required to lift 550 pounds by one foot in one second, is equivalent to about 746 watts. Other units include a logarithmic measure relative to a reference of 1 milliwatt. Power, as a function of time, is the rate at which work is done, so can be expressed by this equation: P = d W d t where P is power, W is work, t is time; because work is a force F applied over a distance x, W = F ⋅ x for a constant force, power can be rewritten as: P = d W d t = d d t = F ⋅ d x d t = F ⋅ v In fact, this is valid for any force, as a consequence of applying the fundamental theorem of calculus.
As a simple example, burning one kilogram of coal releases much more energy than does detonating a kilogram of TNT, but because the TNT reaction releases energy much more it delivers far more power than the coal. If ΔW is the amount of work performed during a period of time of duration Δt, the average power Pavg over that period is given by the formula P a v g = Δ W Δ t, it is the average amount of energy converted per unit of time. The average power is simply called "power" when the context makes it clear; the instantaneous power is the limiting value of the average power as the time interval Δt approaches zero. P = lim Δ t → 0 P a v g = lim Δ t → 0 Δ W Δ t = d W d t. In the case of constant power P, the amount of work performed during a period of duration t is given by: W = P t. In the context of energy conversion, it is more customary to use the symbol E rather than W. Power in mechanical systems is the combination of forces and movement. In particular, power is the product of a force on an object and the object's velocity, or the product of a torque on a shaft and the shaft's angular velocity.
Mechanical power is described as the time derivative of work. In mechanics, the work done by a force F on an object that travels along a curve C is given by the line integral: W C = ∫ C F ⋅ v d t = ∫ C F ⋅ d x, where x defines the path C and v is the velocity along this path. If the force F is derivable from a potential applying the gradi
In analytical mechanics, a branch of theoretical physics, Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the Lagrangian and Hamiltonian functions; the Routhian, like the Hamiltonian, can be obtained from a Legendre transform of the Lagrangian, has a similar mathematical form to the Hamiltonian, but is not the same. The difference between the Lagrangian and Routhian functions are their variables. For a given set of generalized coordinates representing the degrees of freedom in the system, the Lagrangian is a function of the coordinates and velocities, while the Hamiltonian is a function of the coordinates and momenta; the Routhian differs from these functions in that some coordinates are chosen to have corresponding generalized velocities, the rest to have corresponding generalized momenta. This choice is arbitrary, can be done to simplify the problem.
It has the consequence that the Routhian equations are the Hamiltonian equations for some coordinates and corresponding momenta, the Lagrangian equations for the rest of the coordinates and their velocities. In each case the Lagrangian and Hamiltonian functions are replaced by a single function, the Routhian; the full set thus has the advantages of both sets of equations, with the convenience of splitting one set of coordinates to the Hamilton equations, the rest to the Lagrangian equations. The Routhian approach may offer no advantage, but one notable case where this is useful is when a system has cyclic coordinates, by definition those coordinates which do not appear in the original Lagrangian; the Lagrangian equations are powerful results, used in theory and practice, since the equations of motion in the coordinates are easy to set up. However, if cyclic coordinates occur there will still be equations to solve for all the coordinates, including the cyclic coordinates despite their absence in the Lagrangian.
The Hamiltonian equations are useful theoretical results, but less useful in practice because coordinates and momenta are related together in the solutions - after solving the equations the coordinates and momenta must be eliminated from each other. The Hamiltonian equations are suited to cyclic coordinates because the equations in the cyclic coordinates trivially vanish, leaving only the equations in the non cyclic coordinates; the Routhian approach has the best of both approaches, because cyclic coordinates can be split off to the Hamiltonian equations and eliminated, leaving behind the non cyclic coordinates to be solved from the Lagrangian equations. Overall fewer equations need to be solved compared to the Lagrangian approach; as with the rest of analytical mechanics, Routhian mechanics is equivalent to Newtonian mechanics, all other formulations of classical mechanics, introduces no new physics. It offers an alternative way to solve mechanical problems. In the case of Lagrangian mechanics, the generalized coordinates q1, q2... and the corresponding velocities dq1/dt, dq2/dt... and time t, enter the Lagrangian, L, q ˙ i = d q i d t, where the overdots denote time derivatives.
In Hamiltonian mechanics, the generalized coordinates q1, q2... and the corresponding generalized momenta p1, p2... and time, enter the Hamiltonian, H = ∑ i q ˙ i p i − L, p i = ∂ L ∂ q ˙ i, where the second equation is the definition of the generalized momentum pi corresponding to the coordinate qi. The velocities dqi/dt are expressed as functions of their corresponding momenta by inverting their defining relation. In this context, pi is said to be the momentum "canonically conjugate" to qi; the Routhian is intermediate between L and H. The choice of
The velocity of an object is the rate of change of its position with respect to a frame of reference, is a function of time. Velocity is equivalent to a specification of an object's direction of motion. Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity is a physical vector quantity; the scalar absolute value of velocity is called speed, being a coherent derived unit whose quantity is measured in the SI as metres per second or as the SI base unit of. For example, "5 metres per second" is a scalar. If there is a change in speed, direction or both the object has a changing velocity and is said to be undergoing an acceleration. To have a constant velocity, an object must have a constant speed in a constant direction. Constant direction constrains the object to motion in a straight path thus, a constant velocity means motion in a straight line at a constant speed. For example, a car moving at a constant 20 kilometres per hour in a circular path has a constant speed, but does not have a constant velocity because its direction changes.
Hence, the car is considered to be undergoing an acceleration. Speed describes only how fast an object is moving, whereas velocity gives both how fast it is and in which direction the object is moving. If a car is said to travel at 60 km/h, its speed has been specified. However, if the car is said to move at 60 km/h to the north, its velocity has now been specified; the big difference can be noticed. When something moves in a circular path and returns to its starting point, its average velocity is zero but its average speed is found by dividing the circumference of the circle by the time taken to move around the circle; this is because the average velocity is calculated by only considering the displacement between the starting and the end points while the average speed considers only the total distance traveled. Velocity is defined as the rate of change of position with respect to time, which may be referred to as the instantaneous velocity to emphasize the distinction from the average velocity.
In some applications the "average velocity" of an object might be needed, to say, the constant velocity that would provide the same resultant displacement as a variable velocity in the same time interval, v, over some time period Δt. Average velocity can be calculated as: v ¯ = Δ x Δ t; the average velocity is always equal to the average speed of an object. This can be seen by realizing that while distance is always increasing, displacement can increase or decrease in magnitude as well as change direction. In terms of a displacement-time graph, the instantaneous velocity can be thought of as the slope of the tangent line to the curve at any point, the average velocity as the slope of the secant line between two points with t coordinates equal to the boundaries of the time period for the average velocity; the average velocity is the same as the velocity averaged over time –, to say, its time-weighted average, which may be calculated as the time integral of the velocity: v ¯ = 1 t 1 − t 0 ∫ t 0 t 1 v d t, where we may identify Δ x = ∫ t 0 t 1 v d t and Δ t = t 1 − t 0.
If we consider v as velocity and x as the displacement vector we can express the velocity of a particle or object, at any particular time t, as the derivative of the position with respect to time: v = lim Δ t → 0 Δ x Δ t = d x d t. From this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time is the displacement, x. In calculus terms, the integral of the velocity function v is the displacement function x. In the figure, this corresponds to the yellow area under the curve labeled s. X = ∫ v d t. Since the derivative of the position with respect to time gives the change in position divided by the change in time, velocity is measured in metres per second. Although the concept of an instantaneous velocity might at first seem counter-intuitive, it