What is Power and work of a force? — Physics

Definition 13.1 University Physics — Year 1 · Chapter 13 — Work, Energy, and Potential Energy

The power of a force $\vect F$ acting on a point moving at velocity $\vect v$ is

\mathcal P = \vect F\cdot\vect v \qquad (\text{watts}),

and its work between times $t_1$ and $t_2$ (positions $A$ and $B$ ) is the accumulated power,

W_{A\to B} = \int_{t_1}^{t_2}\vect F\cdot\vect v\,\dd t = \int_A^B \vect F\cdot\dd\vect\ell \qquad (\text{joules}),

the sum of the elementary works $\delta W = \vect F\cdot\dd\vect\ell$ along the elementary displacements $\dd\vect\ell = \vect v\,\dd t$ of the trajectory. Work is positive if the force pushes along the motion (motive), negative if against it (resistive), zero if perpendicular.

The elementary work of a force along a small displacement of its point of application: only the component of F along the motion works. Summed along the path from A to B, it gives W_A B. — The elementary work of a force along a small displacement of its point of application: only the component of $\vect F$ along the motion works. Summed along the path from $A$ to $B$ , it gives $W_{A\to B}$ .

Examples

Example 13.10 (The loop’s speed law)

On the frictionless loop of Problem 11.1 the track’s reaction is normal to the motion and works not: $E_m$ is conserved. With $z = R(1 - \cos\theta)$ above the bottom, $\tfrac12 mv^2 + mgR(1 - \cos\theta) = \tfrac12 mv_0^2$ : the speed law $v^2 = v_0^2 - 2gR(1 - \cos\theta)$ used there, now derived in one line. The minimum entry speed $\sqrt{5gR}$ then follows from the top condition $v^2 \geq gR$ of Example 12.16.

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