11.1 Work

11.1.1 Definition of conservative force

Work vectors

  • Work: \[\begin{align} W = \vec{F} \cdot \vec{d} \\ W = \vec{M} \cdot \vec{\theta} \\ \end{align}\]

Notes:

  • Only the component of the force (or moment) in the direction of motion does work.
  • Work is scalar.

In more detail, the equations are: \[\begin{align} W =& \; \vec{F} \cdot \vec{d} \\ W =& \; (F_x i + F_y j + F_z k) \cdot (d_x i + d_y j + d_z k) \\ W =& \; F_x d_x + F_y d_y + F_z d_z \\ \end{align}\]

\[\begin{align} W =& \; \vec{M} \cdot \vec{\theta} \\ W =& \; M_x \theta_x + M_y \theta_y + M_z \theta_z \\ \end{align}\]

11.1.2 Definition of a potential field

Potential (\(V\)) is a function for which: \[\begin{equation*} F_i = - V_{,i} \end{equation*}\] ie: \[\begin{align} F_x = -{\frac{\partial V}{\partial x}} \\ F_y = -{\frac{\partial V}{\partial y}} \\ F_z = -{\frac{\partial V}{\partial z}} \\ \end{align}\]

You are already familiar with potential:

Gravitational Potential

What is a potential function for this scenario?

\[\begin{equation*} V = -m g y + C \end{equation*}\] \[\begin{equation*} {\frac{\partial V}{\partial y}} = - F \end{equation*}\]

\[\begin{equation*} {\frac{\partial V}{\partial y}} = - m g \end{equation*}\]

\[\begin{equation*} F = m g \end{equation*}\]

For a displacement increment \[\begin{equation*} ds = d_x i + d_y j + d_z k \end{equation*}\] the work increment by the force \(F\) is:

\[\begin{align} dW =&\; F \cdot ds = Fx dx + Fy dy + Fz dz \\ dW =&\; -{\frac{\partial V}{\partial x}} dx -{\frac{\partial V}{\partial y}} dy -{\frac{\partial V}{\partial z}} dz \\ dW =&\; - dV \\ \end{align}\]

Therefore: \[\begin{equation*} dW + dV = 0 \end{equation*}\]

11.1.3 Thoughts on potential

\[\begin{equation*} dW + dV = 0 \end{equation*}\]

  • In closed conservative systems (no dissipation) the potential change (\(dV=\delta V=\Delta V\)) is equal to the negative of the work done.
  • Note that in a free fall of an object, its potential decreases by \(mgh\), where \(h\) is the drop in height.
  • The work done by the gravity force \(mg\) during that fall is equal \(mgh\).

11.1.4 External work

Work of external loads Load-Displacement work increment External work definition

The work done by P is: \[\begin{equation*} W_e=\int_0^y P dy \end{equation*}\]

What happens to the energy that goes into the work?

  • The force is transmitted through the bar and causes deformation (strain) of the bar.