The Requirements of Equilibrium — Question 5

Question 5

Three noncollinear moment tests can replace planar force tests

A planar force system has zero net moment about $A=(0,0)$, $B=(2,0)$ and $C=(0,3)\,\mathrm m$. No force acts out of the plane.

1. Prove that its force resultant is zero.

2. Explain why three collinear test points would not give the same guarantee.

Question 5 - Solution

Three noncollinear moment tests can replace planar force tests

(a) Derivation and calculation

The shifts from $A$ to the other two origins give $$\tau_B-\tau_A=-2F_y=0,\qquad \tau_C-\tau_A=+3F_x=0.$$ Hence $$\boxed{F_x=F_y=0}.$$ Together with $\tau_A=0$, these are the three planar equilibrium conditions.

(b) Result and interpretation

For collinear origins with separation $\Delta\vec r=s\hat t$, $$\Delta\tau=-s\hat t\times\vec F_{\mathrm{net}}.$$ Any nonzero resultant parallel to $\hat t$ gives $$\hat t\times\vec F_{\mathrm{net}}=0$$ for every such origin. Noncollinearity is what supplies two independent force constraints.

Physical check

The conclusion applies to the stipulated planar system; extra spatial components would require additional tests.