Chapter 12, Problem 57P

Prove the statement in Section 12.1 that the choice of pivot point doesn't matter when applying conditions for static equilibrium. Figure 12.28 shows an object on which the net force is assumed to be zero. The net torque about the point O is also zero. Show that the net torque about any other point P is also zero. To do so, write the net torque about P as ${\stackrel{\to }{\tau }}_P=\sum {\stackrel{\to }{r}}_{Pi}\times {\stackrel{\to }{F}}_i$ where the vectors ${\stackrel{\to }{r}}_P$ go from P to the force-application points, and the index i labels the different forces. In Fig. 12.28, note that ${\stackrel{\to }{r}}_{Pi}={\stackrel{\to }{r}}_{Oi}\times \stackrel{\to }{R}$where $\stackrel{\to }{R}$ is a vector from P to O. Use this result in your expression for ${\stackrel{\to }{\tau }}_P$ and apply the distributive law to get two separate sums. Use the assumptions that ${\stackrel{\to }{F}}_{net}=\stackrel{\to }{0}$ and ${\stackrel{\to }{\tau }}_O=\stackrel{\to }{0}$ to argue that both terms are zero. This completes the proof.

  

FIGURE 12.28 Problem 51