Work (physics)

In classical mechanics, a force is said to do work when there is a displacement of the center of mass of the body on which the force is applied, in the direction of said force. The work of the force on that body will be equivalent to the energy necessary to displace it. Consequently, it is said that a certain mass has energy when that mass has the capacity to produce work; Furthermore, with this statement it follows that there is no work without energy. For this reason, it is said that coal, gasoline, electricity, and atoms are sources of energy, since they can produce some work or be converted into another type of energy; To understand this, we take into account the universal principle of energy according to which "energy is something that we transform."

In conservative systems, mechanical energy is conserved. If frictional forces are considered, part of the energy is dissipated for example in the form of heat due to the work of the frictional forces.

Work is a scaled physical magnitude that is represented with the letter W{displaystyle W} English Work) and is expressed in energy units, this is in July or joules (J) in the International Unit System.

Since work is by definition a transit of energy, it should never be referred to as increment of work, nor should it be symbolized as "ΔW".

Work in mechanics

I work a force.

Consider a particle P{displaystyle P} on which a force acts F{displaystyle F}, function of particle position in space, this is F=F(r){displaystyle F=F(mathbf {r}}}} and be dr{displaystyle mathrm {d} mathbf {r} } an elementary (infinitesimal) displacement experienced by the particle during a time interval dt{displaystyle mathrm {d} t}. We call elementary work, dW{displaystyle mathrm {d} W}Of strength F{displaystyle mathbf {F} } during elementary displacement dr{displaystyle mathrm {d} mathbf {r} } to product scale F⋅ ⋅ dr{displaystyle Fcdot mathrm {d} mathbf {r} }; this is,

dW=F⋅ ⋅ dr{displaystyle mathrm {d} W=mathbf {F} cdot mathrm {d} mathbf {r} }

If we represent by ds{displaystyle mathrm {d}s} arc length (measured on the trajectory of the particle) in the elemental displacement, this is ds=日本語dr日本語{displaystyle mathrm {d} s=associatedmathrm {d} mathbf {r}{br}} then the tangent vector to the trajectory is given by et=drds{displaystyle mathbf {e} _{text{t}}={frac {mathrm {d} mathbf {r}{mathrm {d}}}}}}}}}{mathrm {d}}}}} and we can write the previous expression in the form

dW=F⋅ ⋅ dr=F⋅ ⋅ etds=(F# θ θ )ds=Fsds{displaystyle mathrm {d} W=mathbf {F} cdot mathrm {d} mathbf {r} =mathbf {F} cdot mathbf {e} _{text{t}}}{mathrm {d} s=(Fcos theta)mathrm {d}{F}{F}{mathrm}{

where θ θ {displaystyle theta } represents the angle determined by the vectors dF{displaystyle mathrm {d} mathbf {F} } and et{displaystyle mathbf {e} and Fs{displaystyle F_{text{s}}} is the force component F in the direction of elementary displacement dr{displaystyle mathrm {d} mathbf {r} }.

The work done by force F{displaystyle mathbf {F} } during an elemental displacement of the particle on which it is applied is a scale, which may be positive, null or negative, depending on the angle θ θ {displaystyle theta } either acute or obtuse.

If the P particle runs a certain trajectory in space, its total displacement between two A and B positions can be considered as the result of adding infinite elemental displacements dr{displaystyle mathrm {d} mathbf {r} } and the total work performed by force F{displaystyle mathbf {F} } in that displacement will be the sum of all these elementary works; that is,

WAB=∫ ∫ ABF⋅ ⋅ dr{displaystyle W_{text{AB}}=int _{text{A}}{text{B}}}{mathbf {F} cdot mathrm {d} mathbf {r}}}} }

This is, the work is given by the curviline integral F{displaystyle mathbf {F} } along the curve C{displaystyle C} which unites the two points; in other words, by the circulation of F{displaystyle mathbf {F} } on the curve C{displaystyle C} between points A and B. Thus, work is a physical magnitude to scale that will generally depend on the trajectory that a point A and B, unless the force F{displaystyle mathbf {F} } be conservative, in which case the work will be independent of the path followed to go from point A to point B, being null in a closed trajectory. Thus, we can say that work is not a state variable.

Particular cases

Constant force on a particle

In the particular case that the force applied to the particle is constant (in magnitude, direction and sense), we have that

WAB=∫ ∫ ABF⋅ ⋅ dr=F⋅ ⋅ ∫ ∫ ABdr=F⋅ ⋅ Δ Δ r=Fs# θ θ {displaystyle W_{text{AB}}=int _{text{A}}{{{text{B}}}}mathbf {F} cdot mathrm {d} mathbf {r} =mathbf {F} cdot int _{text{A}{bc}{mathrm}

that is, the work done by a constant force is expressed by the scalar product of the force times the total displacement vector between the initial and final position. When the force vector is perpendicular to the displacement vector of the body on which it is applied, said force does not do any work. Likewise, if there is no displacement, the work will also be zero.

If a particle operates several forces and we want to calculate the total work done on it, then F{displaystyle mathbf {F} } it will represent the vector resulting from all applied forces.

Work on a rigid solid

In the case of a solid, the total work on it is calculated by adding the contributions on all particles. Mathematically, this work can be expressed as an integral:

W=∫ ∫ VdV∫ ∫ T0TffV(x)⋅ ⋅ v(x)dt{displaystyle W=int _{V}mathrm {d} Vint _{T_{0}}}^{T_{f}}mathbf {f} _{V}(mathbf {x})cdot mathbf {v} (mathbf {x}mathrm {d}{d}}}

If it is a rigid solid the volume forces fV{displaystyle scriptstyle mathbf {f} _{V}} can be written in terms of the resulting force FR{displaystyle scriptstyle mathbf {F} _{R}}the resulting time MR{displaystyle scriptstyle mathbf {M} _{R}}the speed of the mass center VCM{displaystyle scriptstyle mathbf {V} _{CM}} and angular speed ω ω {displaystyle scriptstyle {boldsymbol {omega }}}}:

W=∫ ∫ T0Tf(FR⋅ ⋅ vCM+MR⋅ ⋅ ω ω )dt{displaystyle W=int _{T_{0}}{T_{f}}}}left(mathbf {F} _{R}cdot mathbf {v} _{CM}+mathbf {M} _{R}cdot {boldsymbol {omega}{right}}mathrm {d}

Work and kinetic energy

For the case of a particle in both classical and relativistic mechanics, the following expression is valid:

F=dpdt{displaystyle mathbf {F} ={frac {mathrm {d} mathbf {p} }{mathrm {d}}}}}}

Multiplying this expression scalarly by the velocity and integrating with respect to time, it is obtained that the work done on a particle (classical or relativistic) equals the variation of kinetic energy:

W=∫ ∫ F⋅ ⋅ vdt=∫ ∫ F⋅ ⋅ dr=∫ ∫ v⋅ ⋅ dp=Δ Δ Ec{displaystyle W=int mathbf {F} cdot mathbf {v} mathrm {d} t=int mathbf {F} cdot mathrm {d} mathbf {r} =int mathbf {v} cdot mathrm {d}{d mathbf}{r}{r}}}{r}} =dethbf}} {

This expression is valid in both classical and relativistic mechanics, although given the different relationship between momentum and velocity in both theories, the expression in terms of velocity is slightly different:

{p=mv,classicp=mv1− − v2/c2,mec. relativist⇒ ⇒ ∫ ∫ v⋅ ⋅ dp=Δ Δ Ec={12mv2,classicmc21− − v2/c2,mec. relativist{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFFFF00}}{cHFFFFFFFFFFFF}{cHFFFFFFFF}{cHFFFFFFFF}{cHFFFFFFFFFFFFFFFF}{cH00}{cH00}{cH00}{cH00}{cH00}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00}{cH00}{cH00}{cH00}{cH00}{cH00}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00}{cH00}{cH00}{cH00}{cH00}{cH

Work in thermodynamic energy

In the case of a thermodynamic system, the work is not necessarily of a purely mechanical nature, since the energy exchanged in the interactions can also be calorific, electrical, magnetic or chemical, so it cannot always be expressed in the form of mechanical work.

However, there is a particularly simple and important situation in which work is associated with volume changes experienced by a system (e.g., a fluid contained in an enclosure of variable shape).

So, if we consider a fluid that is subject to external pressure pext{displaystyle p_{text{ext}},} which evolves from a state characterized by a volume V1{displaystyle V_{1}} to another with a volume V2{displaystyle V_{2}}the work done will be:

W12=∫ ∫ V1V2pextdV{displaystyle W_{12}=int _{V_{1}}}{V_{2}{text{ext}}}}{,mathrm {d} V}

resulting in positive work (0}" xmlns="http://www.w3.org/1998/Math/MathML">W▪0{displaystyle W 2005} 0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f912b7a62c30a07b7f4a8ededf6c2ab3692cc86" style="vertical-align: -0.338ex; width:6.696ex; height:2.176ex;"/>) if it is an expansion of the system 0}" xmlns="http://www.w3.org/1998/Math/MathML">dV▪0{displaystyle mathrm {d} V/2003/0} 0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35521c54d0899879643d1ec0a0a85f95198a6c9c" style="vertical-align: -0.338ex; width:7.341ex; height:2.176ex;"/> and negative if not, according to the agreement of signs accepted in the thermodynamics. In a quasistatic process and without friction the external pressure (pext{displaystyle p_{text{ext}}}) will be equal in every moment to the pressure (p{displaystyle p}) of the fluid, so that the work exchanged by the system in these processes is expressed Like

W12=∫ ∫ V1V2pdV{displaystyle W_{12}=int _{V_{1}}}{V_{2}p,mathrm {d} V}

From these expressions it can be inferred that pressure behaves as a generalized force, while volume acts as a generalized displacement. Pressure and volume are a pair of conjugate variables.

In the case that the pressure of the system remains constant during the process, the work is given by:

W=∫ ∫ V1V2pdV=p∫ ∫ V1V2dV=p(V2− − V1)=pΔ Δ V{displaystyle W=int _{V_{1}}{V_{2}}}p,mathrm {d} V=pint _{V_{1}}{V_{2}}}}}}mathrm {d} V=p(V_{2}-V_{1})=pDelta V}

The work in the Clapeyron diagrams of a thermodynamic cycle.

Work units

International System of Units

• July or joule (J), work carried out by a newton (N) of force along one metre (m) away.
  • It is the energy unit (and work) of the SI, named in honor of the 19th century English physicist James Prescott Joule.

Technical System of Units

• Kilogram or kilometerkg), work carried out by a kilopodium (kp) of force along one metre (m) away.
  • Equivalence with the SI: 1 kg = 9,81 J

Cegesimal System of Units

• Ergio (erg), work performed with a force dyna (dyn) along a centimeter (cm) distance.
  • Equivalence with SI: 1 erg = 10^-7 J

US System of Traditional Units

• Pie-libra force (ft·lbf), work performed with a pound-force (lbf) of force along a foot (ft) away. It belongs to both the traditional U.S. unit system and the Imperial System.
  • Equivalence with the SI: 1 ft·lbf = 1,355818 J

Imperial system of units

• Pie-poundalft-pdl), work performed by a poundal (pdl) of force along a foot (ft) away. It belongs to Absolute English system of units.
  • Equivalence with SI: 1 ft-pdl = 0.0421401100938048 J (by definition).

Other units

• kilowatt-hourkW·h)
• Vapor horse·hour (CV·h)