Work

Table of Contents

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

  • Formula: \(W = Fd\)
    • where the \(F\) is the magnitude of the applied force.
    • \(d\) is the displacement of the object.
  • More detailed formula: \(W = Fd\ cos(\theta)\)
  • If the work done is inconsistent and you want to get the total amount of work done within a given time period then:
\begin{align*} W = \int_{x_1}^{x_2} F(x)\, dx \end{align*}
  • where \(x_1\) is the first position of the object.
  • where \(x_2\) is the last position of the object.

Deriviation

  • Other Formulas
\begin{align*} & W = \Delta E_k \\ & W = -\Delta E_p \end{align*}
  • Deriviation for general:
\begin{align*} W &= \int_{t_1}^{t_2} Fv\, dt \\ &= \int_{t_1}^{t_2} F \frac{ds}{\cancel{dt}}\, \cancel{dt} \\ &= \int_{C}F\cdot ds \\ &= F\int_{C} ds \\ &= Fs \end{align*}
  • Deriviation with angle θ:
\begin{align*} W &= \int_{C} F\, ds \\ &= Fs\ cos\theta \end{align*}

References

Date: August 18, 2023

Author: Paul Gerald D. Pare

Emacs 29.1 (Org mode 9.6.6)