PlanetPhysics/Work

In [[../MathematicalFoundationsOfQuantumTheories/|Newtonian mechanics]], work is intimately related to force, as is [[../Momentum/|momentum]]. The very definition of force involved time rate of change of momentum. Work is the integral of a differential consisting of the product of force by a differential element of the displacement of its point of application or of the [[../Particle/|particle]] upon which it is acting. Because both force and displacement are [[../Vectors/|vectors]], one must specify clearly what kind of product is involved in the differential of work, $dW$. If $d𝐥$ is the differential displacement (along) its path) of the particle $P$, upon which force $𝐅$ is acting (or to which $𝐅$ is applied), we define the corresponding of work done by $𝐅$ to be their [[../DotProduct/|scalar product]]. Thus by the definition of work, its differential is

$dW=𝐅\cdot d𝐥=F\cos \theta dl$

The total work done by $𝐅$ while the particle upon which it acts moves along its path from any point $P_1$ to any other point $P_2$ on the path (as in figure) is

$W=\underset{P_1}{\overset{P_2}{\int }}dW=\underset{P_1}{\overset{P_2}{\int }}𝐅\cdot d𝐥=\underset{P_1}{\overset{P_2}{\int }}(F_{x}dx+F_{y}dy+F_{z}dz)$

combing with Eq. 1

$W=\underset{P_1}{\overset{P_2}{\int }}F\cos \theta dl=\underset{P_1}{\overset{P_2}{\int }}{F}_{t}dl$

where $F_t$ is the ([[../Vectors/|scalar]]) component of $𝐅$ tangent to the path in the direction of [[../CosmologicalConstant/|motion]]. This is called the line integral of the force along the path of the particle to which it is applied; it is a scalar. Employing $P_1$ and $P_2$ to represent the limits of integration merely means that for the lower limit we substitute whatever values the variables involved may have at point $P_1$, while for the upper limit we substitute their values at $P_2$. It may be noted that (by the definition of work) no work can be done by any centripetal force (=$m{a}_n$) or by any force exerted by any smooth surface which remains at rest in our inertial [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] of coordinates, though these forces may be very large and important in producing [[../Acceleration/|accelerations]] and determining paths.

The units of work commonly are specified in terms of the corresponding units of force and displacement. Thus the foot-pound (ft-lb) is defined to be the work done by a force of one pound acting through a distance of one foot in its own direction. The work done by a force of one dyne acting through a distance of one centimeter in its own direction is called an erg, usually, rather than a dyne-centimeter. Likewise, the newton-meter is called the joule ($=10^7$ ergs).

If a force $𝐅$ acts upon a [[../RigidBody/|rigid body]] which rotates about a fixed axis, being applied at a point distant $r$ from the axis, the differential work done by it on the body during rotation through the differential angle $d\varphi$ is $dW=F_{tr}(rd\varphi )$, since in this case $F_t=F_{tr}$ and $dl=rd\varphi$. But $r{F}_{tr}=N$ is the [[../AbsoluteMagnitude/|magnitude]] of the moment of $𝐅$ about the axis of rotation. Hence the work done by a torque $N$ (whether moment of a force or of a couple) during rotation of a body to which it is applied, is

$W=\int dW=\int Nd\varphi$

The differential work $Nd\varphi$ done by the torque is positive provided $d\varphi$ s in the sense in which $N$ tends to produce rotation. It should be noted that $F_{tr}$ is normal both to $r$ and to the axis of rotation, and is thus tangent to the path of its point of application.

References

[1] Broxon, James W. "[[../Mechanics/|Mechanics]]" New York, Appleton-Century-Crofts., Inc., 1960.

This entry is a derivative of the Public [[../Bijective/|domain]] work [1].