2.2. Work and Power

In mechanics, as will become clearer later (todo add reference), the concept of work is linked to the concept of energy. todo

2.2.1. Work and Power of a Force

Work. The elementary work of a force \(\vec{F}\) applied at point \(P\) that undergoes an elementary displacement \(d \vec{r}_P\) is defined as the dot product between the force and the displacement,

(2.1)\[\delta W := \vec{F} \cdot d \vec{r}_P \ .\]

The work done by the force \(\vec{F}\) applied at point \(P\) moving from point \(A\) to point \(B\) along the path \(\ell_{AB}\) is the sum of all elementary contributions - and hence, in the limit for elementary displacements \(\rightarrow 0\) for continuous variations, the line integral,

(2.2)\[W_{\ell_{AB}} = \int_{\ell_{AB}} \delta W = \int_{\ell_{AB}} \vec{F} \cdot d \vec{r}_{P} \ .\]

In general, the work of a force or a field of forces depends on the path \({\ell}_{AB}\). In cases where the work is independent of the path but depends only on the endpoints, we talk about conservative actions.

Power. The power of the force is defined as the time derivative of the work,

\[P := \frac{\delta W}{dt} = \vec{F} \cdot \frac{d \vec{r}_P}{d t} = \vec{F} \cdot \vec{v}_P \ , \]

and coincides with the dot product between the force and the velocity of the point of application. Be cautious if a force is applied to geometric points rather than material points, such as in the case of a disk rolling without slipping on a surface: at every instant, the (new) material contact point has zero velocity, while the geometric contact point is the projection of the center of the disk and moves with the same velocity, \(v = R \theta\)

2.2.2. Work and Power of a System of Forces

Work. The work of a system of forces is the sum of the works of the individual forces,

\[\delta W = \sum_{n=1}^{N} \delta W_n = \sum_{n=1}^{N} \vec{F}_n \cdot d \vec{r}_n\]

Power. The power of a system of forces is the sum of the powers of the individual forces,

\[P = \sum_{n=1}^{N} P_n = \sum_{n=1}^{N} \vec{F}_n \cdot \vec{v}_n \ .\]

2.2.3. Work and Power of a Couple of Forces

Work. The elementary work of a couple of forces is the sum of the elementary works,

\[\begin{split}\begin{aligned} \delta W & = \vec{F}_1 \cdot d \vec{r}_1 + \vec{F}_2 \cdot d \vec{r}_2 = \\ & = \vec{F}_1 \cdot ( d \vec{r}_1 - d \vec{r}_2 ) = \end{aligned}\end{split}\]

Power. The power of a couple of forces,

\[P = \vec{F}_1 \cdot (\vec{v}_1 - \vec{v}_2)\]

can be rewritten if the points of application perform a rigid motion act (todo verify the definition of motion act and if it should be introduced),

\[\vec{v}_1 - \vec{v}_2 = \vec{\omega} \times (P_1 - P_2) \ ,\]

as

\[\begin{split}\begin{aligned} P & = \vec{F}_1 \cdot (\vec{v}_1 - \vec{v}_2) = \\ & = \vec{F}_1 \cdot \left[ \vec{\omega} \times (P_1 - P_2) \right] = \\ & = \vec{\omega} \cdot \left[ (P_1 - P_2) \times \vec{F}_1\right] = \\ & = \vec{\omega} \cdot \vec{C} \ . \end{aligned}\end{split}\]