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109-398: If the curvature of the universe is hyperbolic or flat , or if dark energy is a positive cosmological constant , the universe will continue expanding forever, and a heat death is expected to occur, with the universe cooling to approach equilibrium at a very low temperature after a long time period. The hypothesis of heat death stems from the ideas of Lord Kelvin who, in the 1850s, took

218-717: A b d d t Φ ( r ( t ) ) d t = Φ ( r ( b ) ) − Φ ( r ( a ) ) = Φ ( x B ) − Φ ( x A ) . {\displaystyle {\begin{aligned}\int _{\gamma }\nabla \Phi (\mathbf {r} )\cdot d\mathbf {r} &=\int _{a}^{b}\nabla \Phi (\mathbf {r} (t))\cdot \mathbf {r} '(t)dt,\\&=\int _{a}^{b}{\frac {d}{dt}}\Phi (\mathbf {r} (t))dt=\Phi (\mathbf {r} (b))-\Phi (\mathbf {r} (a))=\Phi \left(\mathbf {x} _{B}\right)-\Phi \left(\mathbf {x} _{A}\right).\end{aligned}}} For

327-408: A force field ; such a field is described by vectors at every point in space, which is in-turn called a vector field . A conservative vector field can be simply expressed as the gradient of a certain scalar function, called a scalar potential . The potential energy is related to, and can be obtained from, this potential function. There are various types of potential energy, each associated with

436-625: A bow or a catapult) that is deformed under tension or compression (or stressed in formal terminology). It arises as a consequence of a force that tries to restore the object to its original shape, which is most often the electromagnetic force between the atoms and molecules that constitute the object. If the stretch is released, the energy is transformed into kinetic energy . The gravitational potential function, also known as gravitational potential energy , is: U = − G M m r , {\displaystyle U=-{\frac {GMm}{r}},} The negative sign follows

545-488: A very long timescale. However, if the cosmological constant is positive , the temperature will asymptote to a non-zero positive value, and the universe will approach a state of maximum entropy in which no further work is possible. The theory suggests that from the " Big Bang " through the present day, matter and dark matter in the universe are thought to have been concentrated in stars , galaxies , and galaxy clusters , and are presumed to continue to do so well into

654-562: A Universal Tendency in Nature to the Dissipation of Mechanical Energy , in which he outlined the rudiments of the second law of thermodynamics summarized by the view that mechanical motion and the energy used to create that motion will naturally tend to dissipate or run down. The ideas in this paper, in relation to their application to the age of the Sun and the dynamics of the universal operation, attracted

763-661: A body by a force field is obtained from the gradient of the work, or potential, in the direction of the velocity v of the point of application, that is P ( t ) = − ∇ U ⋅ v = F ⋅ v . {\displaystyle P(t)=-{\nabla U}\cdot \mathbf {v} =\mathbf {F} \cdot \mathbf {v} .} Examples of work that can be computed from potential functions are gravity and spring forces. For small height changes, gravitational potential energy can be computed using U g = m g h , {\displaystyle U_{g}=mgh,} where m

872-432: A coordinate-free way as These formulas can be derived from the special case of arc-length parametrization in the following way. The above condition on the parametrisation imply that the arc length s is a differentiable monotonic function of the parameter t , and conversely that t is a monotonic function of s . Moreover, by changing, if needed, s to – s , one may suppose that these functions are increasing and have

981-404: A distance r is given by Coulomb's Law F = 1 4 π ε 0 Q q r 2 r ^ , {\displaystyle \mathbf {F} ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r^{2}}}\mathbf {\hat {r}} ,} where r ^ {\displaystyle \mathbf {\hat {r}} }

1090-439: A general gravitational field is still not known", and "gravitational entropy is difficult to quantify". The analysis considers several possible assumptions that would be needed for estimates and suggests that the observable universe has more entropy than previously thought. This is because the analysis concludes that supermassive black holes are the largest contributor. Lee Smolin goes further: "It has long been known that gravity

1199-419: A given position and its energy at a reference position. From around 1840 scientists sought to define and understand energy and work . The term "potential energy" was coined by William Rankine a Scottish engineer and physicist in 1853 as part of a specific effort to develop terminology. He chose the term as part of the pair "actual" vs "potential" going back to work by Aristotle . In his 1867 discussion of

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1308-455: A particular type of force. For example, the work of an elastic force is called elastic potential energy; work of the gravitational force is called gravitational potential energy; work of the Coulomb force is called electric potential energy ; work of the strong nuclear force or weak nuclear force acting on the baryon charge is called nuclear potential energy; work of intermolecular forces

1417-411: A positive derivative. Using notation of the preceding section and the chain rule , one has and thus, by taking the norm of both sides where the prime denotes differentiation with respect to t . The curvature is the norm of the derivative of T with respect to s . By using the above formula and the chain rule this derivative and its norm can be expressed in terms of γ ′ and γ ″ only, with

1526-445: A potential are also called conservative forces . The work done by a conservative force is W = − Δ U {\displaystyle W=-\Delta U} where Δ U {\displaystyle \Delta U} is the change in the potential energy associated with the force. The negative sign provides the convention that work done against a force field increases potential energy, while work done by

1635-438: A scalar field, the work of those forces along a curve C is computed by evaluating the scalar field at the start point A and the end point B of the curve. This means the work integral does not depend on the path between A and B and is said to be independent of the path. Potential energy U = − U ′( x ) is traditionally defined as the negative of this scalar field so that work by the force field decreases potential energy, that

1744-567: A special form if the force F is related to a scalar field U ′( x ) so that F = ∇ U ′ = ( ∂ U ′ ∂ x , ∂ U ′ ∂ y , ∂ U ′ ∂ z ) . {\displaystyle \mathbf {F} ={\nabla U'}=\left({\frac {\partial U'}{\partial x}},{\frac {\partial U'}{\partial y}},{\frac {\partial U'}{\partial z}}\right).} This means that

1853-401: A twice differentiable plane curve. Here proper means that on the domain of definition of the parametrization, the derivative ⁠ d γ / dt ⁠ is defined, differentiable and nowhere equal to the zero vector. With such a parametrization, the signed curvature is where primes refer to derivatives with respect to t . The curvature κ is thus These can be expressed in

1962-757: Is W = U ( x A ) − U ( x B ) . {\displaystyle W=U(\mathbf {x} _{\text{A}})-U(\mathbf {x} _{\text{B}}).} In this case, the application of the del operator to the work function 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

2071-873: Is a vector of length 1 pointing from M to m and G is the gravitational constant . Let the mass m move at the velocity v then the work of gravity on this mass as it moves from position r ( t 1 ) to r ( t 2 ) is given by W = − ∫ r ( t 1 ) r ( t 2 ) G M m r 3 r ⋅ d r = − ∫ t 1 t 2 G M m r 3 r ⋅ v d t . {\displaystyle W=-\int _{\mathbf {r} (t_{1})}^{\mathbf {r} (t_{2})}{\frac {GMm}{r^{3}}}\mathbf {r} \cdot d\mathbf {r} =-\int _{t_{1}}^{t_{2}}{\frac {GMm}{r^{3}}}\mathbf {r} \cdot \mathbf {v} \,dt.} The position and velocity of

2180-474: Is (assuming 𝜿 ( s ) ≠ 0) and the center of curvature is on the normal to the curve, the center of curvature is the point (In case the curvature is zero, the center of curvature is not located anywhere on the plane R and is often said to be located "at infinity".) If N ( s ) is the unit normal vector obtained from T ( s ) by a counterclockwise rotation of ⁠ π / 2 ⁠ , then with k ( s ) = ± κ ( s ) . The real number k ( s )

2289-466: Is a function of the state a system is in, and is defined relative to that for a particular state. This reference state is not always a real state; it may also be a limit, such as with the distances between all bodies tending to infinity, provided that the energy involved in tending to that limit is finite, such as in the case of inverse-square law forces. Any arbitrary reference state could be used; therefore it can be chosen based on convenience. Typically

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2398-456: Is a vector of length 1 pointing from Q to q and ε 0 is the vacuum permittivity . The work W required to move q from A to any point B in the electrostatic force field is given by the potential function U ( r ) = 1 4 π ε 0 Q q r . {\displaystyle U(r)={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r}}.} The potential energy

2507-402: Is approximately constant, so the weight force of the ball mg is constant. The product of force and displacement gives the work done, which is equal to the gravitational potential energy, thus U g = m g h . {\displaystyle U_{g}=mgh.} The more formal definition is that potential energy is the energy difference between the energy of an object in

2616-763: Is calculated using its velocity, v = ( v x , v y , v z ) , to obtain W = ∫ 0 t F ⋅ v d t = − ∫ 0 t k x v x d t = − ∫ 0 t k x d x d t d t = ∫ x ( t 0 ) x ( t ) k x d x = 1 2 k x 2 {\displaystyle W=\int _{0}^{t}\mathbf {F} \cdot \mathbf {v} \,dt=-\int _{0}^{t}kxv_{x}\,dt=-\int _{0}^{t}kx{\frac {dx}{dt}}dt=\int _{x(t_{0})}^{x(t)}kx\,dx={\frac {1}{2}}kx^{2}} For convenience, consider contact with

2725-450: Is called intermolecular potential energy. Chemical potential energy, such as the energy stored in fossil fuels , is the work of the Coulomb force during rearrangement of configurations of electrons and nuclei in atoms and molecules. Thermal energy usually has two components: the kinetic energy of random motions of particles and the potential energy of their configuration. Forces derivable from

2834-418: Is called the oriented curvature or signed curvature . It depends on both the orientation of the plane (definition of counterclockwise), and the orientation of the curve provided by the parametrization. In fact, the change of variable s → – s provides another arc-length parametrization, and changes the sign of k ( s ) . Let γ ( t ) = ( x ( t ), y ( t )) be a proper parametric representation of

2943-415: Is closely linked with forces . If the work done by a force on a body that moves from A to B does not depend on the path between these points (if the work is done by a conservative force), then the work of this force measured from A assigns a scalar value to every other point in space and defines a scalar potential field. In this case, the force can be defined as the negative of the vector gradient of

3052-414: Is difficult to manipulate and to express in formulas. Therefore, other equivalent definitions have been introduced. Every differentiable curve can be parametrized with respect to arc length . In the case of a plane curve, this means the existence of a parametrization γ ( s ) = ( x ( s ), y ( s )) , where x and y are real-valued differentiable functions whose derivatives satisfy This means that

3161-454: Is done by introducing a parameterized curve γ ( t ) = r ( t ) from γ ( a ) = A to γ ( b ) = B , and computing, ∫ γ ∇ Φ ( r ) ⋅ d r = ∫ a b ∇ Φ ( r ( t ) ) ⋅ r ′ ( t ) d t , = ∫

3270-439: Is equal to the work done against gravity in lifting it. The work done equals the force required to move it upward multiplied with the vertical distance it is moved (remember W = Fd ). The upward force required while moving at a constant velocity is equal to the weight, mg , of an object, so the work done in lifting it through a height h is the product mgh . Thus, when accounting only for mass , gravity , and altitude ,

3379-409: Is evidenced by water in an elevated reservoir or kept behind a dam. If an object falls from one point to another point inside a gravitational field, the force of gravity will do positive work on the object, and the gravitational potential energy will decrease by the same amount. Consider a book placed on top of a table. As the book is raised from the floor to the table, some external force works against

Heat death of the universe - Misplaced Pages Continue

3488-409: Is important for keeping the universe out of thermal equilibrium. Gravitationally bound systems have negative specific heat—that is, the velocities of their components increase when energy is removed. ... Such a system does not evolve toward a homogeneous equilibrium state. Instead it becomes increasingly structured and heterogeneous as it fragments into subsystems." This point of view is also supported by

3597-462: Is measured. Choosing the convention that K = 0 (i.e. in relation to a point at infinity) makes calculations simpler, albeit at the cost of making U negative; for why this is physically reasonable, see below. Given this formula for U , the total potential energy of a system of n bodies is found by summing, for all n ( n − 1 ) 2 {\textstyle {\frac {n(n-1)}{2}}} pairs of two bodies,

3706-409: Is more complex, as it depends on the choice of a direction on the surface or manifold. This leads to the concepts of maximal curvature , minimal curvature , and mean curvature . In Tractatus de configurationibus qualitatum et motuum, the 14th-century philosopher and mathematician Nicole Oresme introduces the concept of curvature as a measure of departure from straightness; for circles he has

3815-635: Is negligible and we can assume that the force of gravity on a particular object is constant. Near the surface of the Earth, for example, we assume that the acceleration due to gravity is a constant g = 9.8 m/s ( standard gravity ). In this case, a simple expression for gravitational potential energy can be derived using the W = Fd equation for work , and the equation W F = − Δ U F . {\displaystyle W_{F}=-\Delta U_{F}.} The amount of gravitational potential energy held by an elevated object

3924-512: Is not in equilibrium, we cannot associate an entropy with it." Hans Adolf Buchdahl writes of "the entirely unjustifiable assumption that the universe can be treated as a closed thermodynamic system". According to Giovanni Gallavotti , "there is no universally accepted notion of entropy for systems out of equilibrium, even when in a stationary state". Discussing the question of entropy for non-equilibrium states in general, Elliott H. Lieb and Jakob Yngvason express their opinion as follows: "Despite

4033-411: Is often given as a definition of the curvature. Historically, the curvature of a differentiable curve was defined through the osculating circle , which is the circle that best approximates the curve at a point. More precisely, given a point P on a curve, every other point Q of the curve defines a circle (or sometimes a line) passing through Q and tangent to the curve at P . The osculating circle

4142-791: Is possible that the universe may enter a second inflationary epoch, or assuming that the current vacuum state is a false vacuum , the vacuum may decay into a lower- energy state . It is also possible that entropy production will cease and the universe will reach heat death. It is suggested that, over vast periods of time, a spontaneous entropy decrease would eventually occur via the Poincaré recurrence theorem , thermal fluctuations , and fluctuation theorem . Through this, another universe could possibly be created by random quantum fluctuations or quantum tunnelling in roughly 10 10 10 56 {\displaystyle 10^{10^{10^{56}}}} years. Max Planck wrote that

4251-496: Is possible with the real number system. Since physicists abhor infinities in their calculations, and r is always non-zero in practice, the choice of U = 0 {\displaystyle U=0} at infinity is by far the more preferable choice, even if the idea of negative energy in a gravity well appears to be peculiar at first. The negative value for gravitational energy also has deeper implications that make it seem more reasonable in cosmological calculations where

4360-399: Is somewhat arbitrary, as it depends on the orientation of the curve. In the case of the graph of a function, there is a natural orientation by increasing values of x . This makes significant the sign of the signed curvature. The sign of the signed curvature is the same as the sign of the second derivative of f . If it is positive then the graph has an upward concavity, and, if it is negative

4469-588: Is still too hot for life to arise there for thousands of years, while the Moon is already too cold. The final state, in this view, is described as one of "equilibrium" in which all motion ceases. The idea of heat death as a consequence of the laws of thermodynamics, however, was first proposed in loose terms beginning in 1851 by Lord Kelvin (William Thomson), who theorized further on the mechanical energy loss views of Sadi Carnot (1824), James Joule (1843) and Rudolf Clausius (1850). Thomson's views were then elaborated over

Heat death of the universe - Misplaced Pages Continue

4578-426: Is the limit , if it exists, of this circle when Q tends to P . Then the center and the radius of curvature of the curve at P are the center and the radius of the osculating circle. The curvature is the reciprocal of radius of curvature. That is, the curvature is where R is the radius of curvature (the whole circle has this curvature, it can be read as turn 2π over the length 2π R ). This definition

4687-464: Is the mass in kilograms, g is the local gravitational field (9.8 metres per second squared on Earth), h is the height above a reference level in metres, and U is the energy in joules. In classical physics, gravity exerts a constant downward force F = (0, 0, F z ) on the center of mass of a body moving near the surface of the Earth. The work of gravity on a body moving along a trajectory r ( t ) = ( x ( t ), y ( t ), z ( t )) , such as

4796-487: Is the trajectory taken from A to B. Because the work done is independent of the path taken, then this expression is true for any trajectory, C , from A to B. The function U ( x ) is called the potential energy associated with the applied force. Examples of forces that have potential energies are gravity and spring forces. In this section the relationship between work and potential energy is presented in more detail. The line integral that defines work along curve C takes

4905-429: 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 = U ( x A ) − U ( x B ) {\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {x} =U(\mathbf {x} _{\text{A}})-U(\mathbf {x} _{\text{B}})} where C

5014-475: The International System of Units (SI) is the joule (symbol J). Potential energy is associated with forces that act on a body in a way that the total work done by these forces on the body depends only on the initial and final positions of the body in space. These forces, whose total work is path independent, are called conservative forces . If the force acting on a body varies over space, then one has

5123-505: The second law of thermodynamics . In Isaac Asimov's 1956 short story The Last Question , humans repeatedly wonder how the heat death of the universe can be avoided. In the 1981 Doctor Who story " Logopolis ", the Doctor realizes that the Logopolitans have created vents in the universe to expel heat build-up into other universes—"Charged Vacuum Emboitments" or "CVE"—to delay the demise of

5232-453: The theory of heat as mechanical energy loss in nature (as embodied in the first two laws of thermodynamics ) and extrapolated it to larger processes on a universal scale. This also allowed Kelvin to formulate the heat death paradox , which disproves an infinitely old universe. The idea of heat death stems from the second law of thermodynamics , of which one version states that entropy tends to increase in an isolated system . From this,

5341-444: The wave equation of a string under tension, and other applications where small slopes are involved. This often allows systems that are otherwise nonlinear to be treated approximately as linear. If a curve is defined in polar coordinates by the radius expressed as a function of the polar angle, that is r is a function of θ , then its curvature is where the prime refers to differentiation with respect to θ . This results from

5450-486: The Entropics as a stand in for the effects of a heat death. Curvature#Space In mathematics , curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane . If a curve or surface is contained in a larger space, curvature can be defined extrinsically relative to

5559-498: The Moon's gravity is weaker. "Height" in the common sense of the term cannot be used for gravitational potential energy calculations when gravity is not assumed to be a constant. The following sections provide more detail. The strength of a gravitational field varies with location. However, when the change of distance is small in relation to the distances from the center of the source of the gravitational field, this variation in field strength

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5668-450: The ambient space. Curvature of Riemannian manifolds of dimension at least two can be defined intrinsically without reference to a larger space. For curves, the canonical example is that of a circle , which has a curvature equal to the reciprocal of its radius . Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle — that is,

5777-466: The antagonist Kyubey reveals he is a member of an alien race who has been creating magical girls for millennia in order to harvest their energy to combat entropy and stave off the heat death of the universe. In the last act of Final Fantasy XIV: Endwalker , the player encounters an alien race known as the Ea who have lost all hope in the future and any desire to live further, all because they have learned of

5886-413: The approximation that g is constant is no longer valid, and we have to use calculus and the general mathematical definition of work to determine gravitational potential energy. For the computation of the potential energy, we can integrate the gravitational force, whose magnitude is given by Newton's law of gravitation , with respect to the distance r between the two bodies. Using that definition,

5995-409: The arc-length parameter s completely eliminated, giving the above formulas for the curvature. The graph of a function y = f ( x ) , is a special case of a parametrized curve, of the form As the first and second derivatives of x are 1 and 0, previous formulas simplify to for the curvature, and to for the signed curvature. In the general case of a curve, the sign of the signed curvature

6104-470: The choice of zero point is arbitrary. Given that there is no reasonable criterion for preferring one particular finite r over another, there seem to be only two reasonable choices for the distance at which U becomes zero: r = 0 {\displaystyle r=0} and r = ∞ {\displaystyle r=\infty } . The choice of U = 0 {\displaystyle U=0} at infinity may seem peculiar, and

6213-413: The circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent , which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number . For surfaces (and, more generally for higher-dimensional manifolds ), that are embedded in a Euclidean space , the concept of curvature

6322-528: The clock's direction, resulting in a "rejuvenating universe" – would require "a creative act or an act possessing similar power". Starting from this publication, Kelvin also introduced the heat death paradox (Kelvin's paradox), which challenged the classical concept of an infinitely old universe, since the universe has not achieved its thermodynamic equilibrium, thus further work and entropy production are still possible. The existence of stars and temperature differences can be considered an empirical proof that

6431-431: The collapse of superclusters of galaxies. Even these would evaporate over a timescale of up to 10 years. After that time, the universe enters the so-called Dark Era and is expected to consist chiefly of a dilute gas of photons and leptons . With only very diffuse matter remaining, activity in the universe will have tailed off dramatically, with extremely low energy levels and extremely long timescales. Speculatively, it

6540-713: The consequence that gravitational energy is always negative may seem counterintuitive, but this choice allows gravitational potential energy values to be finite, albeit negative. The singularity at r = 0 {\displaystyle r=0} in the formula for gravitational potential energy means that the only other apparently reasonable alternative choice of convention, with U = 0 {\displaystyle U=0} for r = 0 {\displaystyle r=0} , would result in potential energy being positive, but infinitely large for all nonzero values of r , and would make calculations involving sums or differences of potential energies beyond what

6649-499: The convention that work is gained from a loss of potential energy. The gravitational force between two bodies of mass M and m separated by a distance r is given by Newton's law of universal gravitation F = − G M m r 2 r ^ , {\displaystyle \mathbf {F} =-{\frac {GMm}{r^{2}}}\mathbf {\hat {r}} ,} where r ^ {\displaystyle \mathbf {\hat {r}} }

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6758-425: The curvature as being inversely proportional to the radius; and he attempts to extend this idea to other curves as a continuously varying magnitude. The curvature of a differentiable curve was originally defined through osculating circles . In this setting, Augustin-Louis Cauchy showed that the center of curvature is the intersection point of two infinitely close normal lines to the curve. curve Intuitively,

6867-423: The curvature can be derived from the expression of the curvature of the graph of a function by using the implicit function theorem and the fact that, on such a curve, one has It can be useful to verify on simple examples that the different formulas given in the preceding sections give the same result. A common parametrization of a circle of radius r is γ ( t ) = ( r cos t , r sin t ) . The formula for

6976-451: The curvature describes for any part of a curve how much the curve direction changes over a small distance travelled (e.g. angle in rad/m ), so it is a measure of the instantaneous rate of change of direction of a point that moves on the curve: the larger the curvature, the larger this rate of change. In other words, the curvature measures how fast the unit tangent vector to the curve at point p rotates when point p moves at unit speed along

7085-441: The curvature gives It follows, as expected, that the radius of curvature is the radius of the circle, and that the center of curvature is the center of the circle. The circle is a rare case where the arc-length parametrization is easy to compute, as it is It is an arc-length parametrization, since the norm of is equal to one. This parametrization gives the same value for the curvature, as it amounts to division by r in both

7194-443: The curve at P ( s ) , which is also the derivative of P ( s ) with respect to s . Then, the derivative of T ( s ) with respect to s is a vector that is normal to the curve and whose length is the curvature. To be meaningful, the definition of the curvature and its different characterizations require that the curve is continuously differentiable near P , for having a tangent that varies continuously; it requires also that

7303-421: The curve defined by F ( x , y ) = 0 , but it would change the sign of the numerator if the absolute value were omitted in the preceding formula. A point of the curve where F x = F y = 0 is a singular point , which means that the curve is not differentiable at this point, and thus that the curvature is not defined (most often, the point is either a crossing point or a cusp ). The above formula for

7412-446: The curve is twice differentiable at P , for insuring the existence of the involved limits, and of the derivative of T ( s ) . The characterization of the curvature in terms of the derivative of the unit tangent vector is probably less intuitive than the definition in terms of the osculating circle, but formulas for computing the curvature are easier to deduce. Therefore, and also because of its use in kinematics , this characterization

7521-400: The curve. In fact, it can be proved that this instantaneous rate of change is exactly the curvature. More precisely, suppose that the point is moving on the curve at a constant speed of one unit, that is, the position of the point P ( s ) is a function of the parameter s , which may be thought as the time or as the arc length from a given origin. Let T ( s ) be a unit tangent vector of

7630-495: The equation is: U = m g h {\displaystyle U=mgh} where U is the potential energy of the object relative to its being on the Earth's surface, m is the mass of the object, g is the acceleration due to gravity, and h is the altitude of the object. Hence, the potential difference is Δ U = m g Δ h . {\displaystyle \Delta U=mg\Delta h.} However, over large variations in distance,

7739-528: The eventual heat death of the universe and see everything else as pointless due to its probable inevitability. The overarching plot of the Xeelee Sequence concerns the Photino Birds' efforts to accelerate the heat death of the universe by accelerating the rate at which stars become white dwarves. The 2019 hit indie video game Outer Wilds has several themes grappling with the idea of the heat death of

7848-412: The fact of a recent experimental discovery of a stable non-equilibrium steady state in a relatively simple closed system. It should be expected that an isolated system fragmented into subsystems does not necessarily come to thermodynamic equilibrium and remain in non-equilibrium steady state. Entropy will be transmitted from one subsystem to another, but its production will be zero, which does not contradict

7957-416: The fact that d d t r − 1 = − r − 2 r ˙ = − r ˙ r 2 . {\displaystyle {\frac {d}{dt}}r^{-1}=-r^{-2}{\dot {r}}=-{\frac {\dot {r}}{r^{2}}}.} The electrostatic force exerted by a charge Q on another charge q separated by

8066-470: The fact that most physicists believe in such a nonequilibrium entropy, it has so far proved impossible to define it in a clearly satisfactory way." In Peter Landsberg's opinion: "The third misconception is that thermodynamics, and in particular, the concept of entropy, can without further enquiry be applied to the whole universe. ... These questions have a certain fascination, but the answers are speculations." A 2010 analysis of entropy states, "The entropy of

8175-438: The final state of the universe depend on the assumptions made about its ultimate fate, and these assumptions have varied considerably over the late 20th century and early 21st century. In a hypothesized "open" or "flat" universe that continues expanding indefinitely, either a heat death or a Big Rip is expected to eventually occur. If the cosmological constant is zero, the universe will approach absolute zero temperature over

8284-423: The force F is said to be "derivable from a potential". This also necessarily implies that F must be a conservative vector field . The potential U defines a force F at every point x in space, so the set of forces is called a force field . Given a force field F ( x ), evaluation of the work integral using the gradient theorem can be used to find the scalar function associated with potential energy. This

8393-797: The force field F , let v = d r / dt , then the gradient theorem yields, ∫ γ F ⋅ d r = ∫ a b F ⋅ v d t , = − ∫ a b d d t U ( r ( t ) ) d t = U ( x A ) − U ( x B ) . {\displaystyle {\begin{aligned}\int _{\gamma }\mathbf {F} \cdot d\mathbf {r} &=\int _{a}^{b}\mathbf {F} \cdot \mathbf {v} \,dt,\\&=-\int _{a}^{b}{\frac {d}{dt}}U(\mathbf {r} (t))\,dt=U(\mathbf {x} _{A})-U(\mathbf {x} _{B}).\end{aligned}}} The power applied to

8502-423: The force field decreases potential energy. Common notations for potential energy are PE , U , V , and E p . Potential energy is the energy by virtue of an object's position relative to other objects. Potential energy is often associated with restoring forces such as a spring or the force of gravity . The action of stretching a spring or lifting a mass is performed by an external force that works against

8611-423: The force field of the potential. This work is stored in the force field, which is said to be stored as potential energy. If the external force is removed the force field acts on the body to perform the work as it moves the body back to the initial position, reducing the stretch of the spring or causing a body to fall. Consider a ball whose mass is m dropped from height h . The acceleration g of free fall

8720-449: The formula for general parametrizations, by considering the parametrization For a curve defined by an implicit equation F ( x , y ) = 0 with partial derivatives denoted F x , F y , F xx , F xy , F yy , the curvature is given by The signed curvature is not defined, as it depends on an orientation of the curve that is not provided by the implicit equation. Note that changing F into – F would not change

8829-440: The future. Therefore, the universe is not in thermodynamic equilibrium , and objects can do physical work. The decay time for a supermassive black hole of roughly 1 galaxy mass (10  solar masses ) because of Hawking radiation is in the order of 10  years, so entropy can be produced until at least that time. Some large black holes in the universe are predicted to continue to grow up to perhaps 10 M ☉ during

8938-461: The graph has a downward concavity. If it is zero, then one has an inflection point or an undulation point . When the slope of the graph (that is the derivative of the function) is small, the signed curvature is well approximated by the second derivative. More precisely, using big O notation , one has It is common in physics and engineering to approximate the curvature with the second derivative, for example, in beam theory or for deriving

9047-524: The gravitational force. If the book falls back to the floor, the "falling" energy the book receives is provided by the gravitational force. Thus, if the book falls off the table, this potential energy goes to accelerate the mass of the book and is converted into kinetic energy . When the book hits the floor this kinetic energy is converted into heat, deformation, and sound by the impact. The factors that affect an object's gravitational potential energy are its height relative to some reference point, its mass, and

9156-469: The gravitational potential energy of a system of masses m 1 and M 2 at a distance r using the Newtonian constant of gravitation G is U = − G m 1 M 2 r + K , {\displaystyle U=-G{\frac {m_{1}M_{2}}{r}}+K,} where K is an arbitrary constant dependent on the choice of datum from which potential

9265-399: The hypothesis implies that if the universe lasts for a sufficient time, it will asymptotically approach a state where all energy is evenly distributed. In other words, according to this hypothesis, there is a tendency in nature towards the dissipation (energy transformation) of mechanical energy (motion) into thermal energy ; hence, by extrapolation, there exists the view that, in time,

9374-400: The integral of the vertical component of velocity is the vertical distance. The work of gravity depends only on the vertical movement of the curve r ( t ) . A horizontal spring exerts a force F = (− kx , 0, 0) that is proportional to its deformation in the axial or x direction. The work of this spring on a body moving along the space curve s ( t ) = ( x ( t ), y ( t ), z ( t )) ,

9483-532: The likes of William Rankine and Hermann von Helmholtz. The three of them were said to have exchanged ideas on this subject. In 1862, Thomson published "On the age of the Sun's heat", an article in which he reiterated his fundamental beliefs in the indestructibility of energy (the first law ) and the universal dissipation of energy (the second law), leading to diffusion of heat, cessation of useful motion ( work ), and exhaustion of potential energy , "lost irrecoverably" through

9592-401: The mass m are given by r = r e r , v = r ˙ e r + r θ ˙ e t , {\displaystyle \mathbf {r} =r\mathbf {e} _{r},\qquad \mathbf {v} ={\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{t},} where e r and e t are

9701-430: The material universe, while clarifying his view of the consequences for the universe as a whole. Thomson wrote: The result would inevitably be a state of universal rest and death, if the universe were finite and left to obey existing laws. But it is impossible to conceive a limit to the extent of matter in the universe; and therefore science points rather to an endless progress, through an endless space, of action involving

9810-627: The mechanical movement of the universe will run down as work is converted to heat because of the second law. The conjecture that all bodies in the universe cool off, eventually becoming too cold to support life, seems to have been first put forward by the French astronomer Jean Sylvain Bailly in 1777 in his writings on the history of astronomy and in the ensuing correspondence with Voltaire . In Bailly's view, all planets have an internal heat and are now at some particular stage of cooling. Jupiter , for instance,

9919-553: The next decade by Hermann von Helmholtz and William Rankine . The idea of the heat death of the universe derives from discussion of the application of the first two laws of thermodynamics to universal processes. Specifically, in 1851, Lord Kelvin outlined the view, as based on recent experiments on the dynamical theory of heat : "heat is not a substance, but a dynamical form of mechanical effect, we perceive that there must be an equivalence between mechanical work and heat, as between cause and effect." In 1852, Thomson published On

10028-478: The numerator and the denominator in the preceding formula. Potential energy U = 1 ⁄ 2 ⋅ k ⋅ x ( elastic ) U = 1 ⁄ 2 ⋅ C ⋅ V ( electric ) U = − m ⋅ B ( magnetic ) In physics , potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. The term potential energy

10137-449: The phrase "entropy of the universe" has no meaning because it admits of no accurate definition. In 2008, Walter Grandy wrote: "It is rather presumptuous to speak of the entropy of a universe about which we still understand so little, and we wonder how one might define thermodynamic entropy for a universe and its major constituents that have never been in equilibrium in their entire existence." According to László Tisza , "If an isolated system

10246-412: The potential energy of a system depends on the relative positions of its components only, so the reference state can also be expressed in terms of relative positions. Gravitational energy is the potential energy associated with gravitational force , as work is required to elevate objects against Earth's gravity. The potential energy due to elevated positions is called gravitational potential energy, and

10355-467: The potential energy of the system of those two bodies. Considering the system of bodies as the combined set of small particles the bodies consist of, and applying the previous on the particle level we get the negative gravitational binding energy . This potential energy is more strongly negative than the total potential energy of the system of bodies as such since it also includes the negative gravitational binding energy of each body. The potential energy of

10464-401: The potential field. If the work for an applied force is independent of the path, then the work done by the force is evaluated from the start to the end of the trajectory of the point of application. This means that there is a function U ( x ), called a "potential", that can be evaluated at the two points x A and x B to obtain the work over any trajectory between these two points. It

10573-1090: The radial and tangential unit vectors directed relative to the vector from M to m . Use this to simplify the formula for work of gravity to, W = − ∫ t 1 t 2 G m M r 3 ( r e r ) ⋅ ( r ˙ e r + r θ ˙ e t ) d t = − ∫ t 1 t 2 G m M r 3 r r ˙ d t = G M m r ( t 2 ) − G M m r ( t 1 ) . {\displaystyle W=-\int _{t_{1}}^{t_{2}}{\frac {GmM}{r^{3}}}(r\mathbf {e} _{r})\cdot ({\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{t})\,dt=-\int _{t_{1}}^{t_{2}}{\frac {GmM}{r^{3}}}r{\dot {r}}dt={\frac {GMm}{r(t_{2})}}-{\frac {GMm}{r(t_{1})}}.} This calculation uses

10682-421: The same topic Rankine describes potential energy as ‘energy of configuration’ in contrast to actual energy as 'energy of activity'. Also in 1867, William Thomson introduced "kinetic energy" as the opposite of "potential energy", asserting that all actual energy took the form of ⁠ 1 / 2 ⁠ mv . Once this hypothesis became widely accepted, the term "actual energy" gradually faded. Potential energy

10791-418: The spring occurs at t = 0 , then the integral of the product of the distance x and the x -velocity, xv x , is x /2. The function U ( x ) = 1 2 k x 2 , {\displaystyle U(x)={\frac {1}{2}}kx^{2},} is called the potential energy of a linear spring. Elastic potential energy is the potential energy of an elastic object (for example

10900-404: The strength of the gravitational field it is in. Thus, a book lying on a table has less gravitational potential energy than the same book on top of a taller cupboard and less gravitational potential energy than a heavier book lying on the same table. An object at a certain height above the Moon's surface has less gravitational potential energy than at the same height above the Earth's surface because

11009-755: The system of bodies as such is the negative of the energy needed to separate the bodies from each other to infinity, while the gravitational binding energy is the energy needed to separate all particles from each other to infinity. U = − m ( G M 1 r 1 + G M 2 r 2 ) {\displaystyle U=-m\left(G{\frac {M_{1}}{r_{1}}}+G{\frac {M_{2}}{r_{2}}}\right)} therefore, U = − m ∑ G M r , {\displaystyle U=-m\sum G{\frac {M}{r}},} As with all potential energies, only differences in gravitational potential energy matter for most physical purposes, and

11118-402: The tangent vector has a length equal to one and is thus a unit tangent vector . If the curve is twice differentiable, that is, if the second derivatives of x and y exist, then the derivative of T ( s ) exists. This vector is normal to the curve, its length is the curvature κ ( s ) , and it is oriented toward the center of curvature. That is, Moreover, because the radius of curvature

11227-552: The track of a roller coaster is calculated using its velocity, v = ( v x , v y , v z ) , to obtain W = ∫ t 1 t 2 F ⋅ v d t = ∫ t 1 t 2 F z v z d t = F z Δ z . {\displaystyle W=\int _{t_{1}}^{t_{2}}{\boldsymbol {F}}\cdot {\boldsymbol {v}}\,dt=\int _{t_{1}}^{t_{2}}F_{z}v_{z}\,dt=F_{z}\Delta z.} where

11336-425: The transformation of potential energy into palpable motion and hence into heat , than to a single finite mechanism, running down like a clock, and stopping for ever. The clock's example shows how Kelvin was unsure whether the universe would eventually achieve thermodynamic equilibrium . Thompson later speculated that restoring the dissipated energy in " vis viva " and then usable work – and therefore revert

11445-698: The units of U ′ must be this case, work along the curve is given by W = ∫ C F ⋅ d x = ∫ C ∇ U ′ ⋅ d x , {\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {x} =\int _{C}\nabla U'\cdot d\mathbf {x} ,} which can be evaluated using the gradient theorem to obtain W = U ′ ( x B ) − U ′ ( x A ) . {\displaystyle W=U'(\mathbf {x} _{\text{B}})-U'(\mathbf {x} _{\text{A}}).} This shows that when forces are derivable from

11554-406: The universe is not infinitely old. In the years to follow both Thomson's 1852 and the 1862 papers, Helmholtz and Rankine both credited Thomson with the idea, along with his paradox, but read further into his papers by publishing views stating that Thomson argued that the universe will end in a "heat death" (Helmholtz), which will be the "end of all physical phenomena" (Rankine). Proposals about

11663-424: The universe, and the theory that the universe is a cycle of big bangs once the previous one has experienced a heat death. In Singularity Immemorial — the 7th main story event of a mobile game Girls' Frontline: Neural Cloud — the plot is about a virtual sector made to simulate space exploration and the threat of the heat death of the universe. The simulation uses an imitation of Neural Cloud's virus entities known as

11772-483: The universe. The Doctor unwittingly travelled through such a vent in " Full Circle ". In the 1995 computer game I Have No Mouth, and I Must Scream , based on the Harlan Ellison short story of the same name , it is stated that AM, the malevolent supercomputer, will survive the heat death of the universe and continue torturing its immortal victims to eternity. In the 2011 anime series Puella Magi Madoka Magica ,

11881-440: Was introduced by the 19th-century Scottish engineer and physicist William Rankine , although it has links to the ancient Greek philosopher Aristotle 's concept of potentiality . Common types of potential energy include the gravitational potential energy of an object, the elastic potential energy of a deformed spring, and the electric potential energy of an electric charge in an electric field . The unit for energy in

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