Work

Definition

action done on an object that displaces the object.
Work is the dot product of force and displacement. It is a scalar quantity.
Work can be ± and if it is -, then the work done is opposite of the direction of the displacement.
it can also be defined as a change in energy.
Work is done only when a force succeeds in moving the body upon which it acts.
- Work is only done if displacement occurs.
The SI Unit of work is Joules (J) which is in $N \cdot m$ .
only the component of force parallel to the displacement counts towards work.

Formula: $W = F d$
- where the $F$ is the magnitude of the applied force.
- $d$ is the displacement of the object.
More detailed formula: $W = F d c o s (θ)$
If the work done is inconsistent and you want to get the total amount of work done within a given time period then:

\begin{array}{r} W = \int_{x_{1}}^{x_{2}} F (x) d x \end{array}

\begin{aligned} W = Δ E_{k} \\ W = - Δ E_{p} \end{aligned}

\begin{aligned} W & = \int_{t_{1}}^{t_{2}} F v d t \\ = \int_{t_{1}}^{t_{2}} F \frac{d s}{d t} d t \\ = \int_{C} F \cdot d s \\ = F \int_{C} d s \\ = F s \end{aligned}

\begin{aligned} W & = \int_{C} F d s \\ = F s c o s θ \end{aligned}