Dynamics of extended bodies

Torque = force \(\times\) lever arm, $\tau=F \, l$

  • sign of $\tau$ = sense of rotation (+ve c.c.w.)
  • lever arm = shortest distance to the line of force

Equilibrium: $\sum \vec{F} = 0$ and $\sum \tau = 0$

  • provides another source of equations
  • the point about which $\sum \tau = 0$ is arbitrary

Center of gravity

  • c.o.g. = the point about which total \(\tau_{\rm weight}=0\)
  • In problems involving weight: $\vec{W}$ acts on the body at c.o.g., and thus causes no torque about c.o.g.

N2L for rotations \[ \tau = I \, \alpha, \quad \mbox{or} \quad \sum \tau = I_{\rm body} \alpha \] where the moment of inertia \(I\) is a property of the extended body, determined by the spatial distribution of mass: \[ I=\sum_i m_i r_i^2 \]