Talk:PlanetPhysics/Work

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In \htmladdnormallink{Newtonian mechanics}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html}, work is intimately related to force, as is \htmladdnormallink{momentum}{http://planetphysics.us/encyclopedia/Momentum.html}. 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 \htmladdnormallink{particle}{http://planetphysics.us/encyclopedia/Particle.html} upon which it is acting. Because both force and displacement are \htmladdnormallink{vectors}{http://planetphysics.us/encyclopedia/Vectors.html}, one must specify clearly what kind of product is involved in the differential of work, $dW$. If $d \mathbf{l}$ is the differential displacement (along) its path) of the particle $P$, upon which force $\mathbf{F}$ is acting (or to which $\mathbf{F}$ is applied), we define the corresponding of work done by $\mathbf{F}$ to be their \htmladdnormallink{scalar product}{http://planetphysics.us/encyclopedia/DotProduct.html}. Thus by the definition of work, its differential is

\begin{equation} dW = \mathbf{F} \cdot d\mathbf{l} = F \cos \theta dl \end{equation}

\begin{center} \includegraphics[scale=0.6]{work.eps} \end{center}

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

$$ W = \int_{P_1}^{P_2} dW = \int_{P_1}^{P_2} \mathbf{F} \cdot d \mathbf{l} = \int_{P_1}^{P_2} \left( F_x dx + F_y dy +F_z dz \right ) $$

combing with Eq. 1

\begin{equation} W = \int_{P_1}^{P_2} F \cos \theta dl = \int_{P_1}^{P_2} F_t dl \end{equation}

where $F_t$ is the (\htmladdnormallink{scalar}{http://planetphysics.us/encyclopedia/Vectors.html}) component of $\mathbf{F}$ tangent to the path in the direction of \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}. 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 (=$ma_n$) or by any force exerted by any smooth surface which remains at rest in our inertial \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} of coordinates, though these forces may be very large and important in producing \htmladdnormallink{accelerations}{http://planetphysics.us/encyclopedia/Acceleration.html} 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 $\mathbf{F}$ acts upon a \htmladdnormallink{rigid body}{http://planetphysics.us/encyclopedia/RigidBody.html} 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 $d \phi$ is $dW = F_{tr} ( r d\phi)$, since in this case $F_t = F_{tr}$ and $dl = r d\phi$. But $rF_{tr} = N$ is the \htmladdnormallink{magnitude}{http://planetphysics.us/encyclopedia/AbsoluteMagnitude.html} of the moment of $\mathbf{F}$ 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

\begin{equation} W = \int dW = \int N d\phi \end{equation}

The differential work $N d\phi$ done by the torque is positive provided $d \phi$ 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.

\subsection{References}

[1] Broxon, James W. "\htmladdnormallink{Mechanics}{http://planetphysics.us/encyclopedia/Mechanics.html}" New York, Appleton-Century-Crofts., Inc., 1960.

This entry is a derivative of the Public \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} work [1].

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