The Requirements of Equilibrium — Question 4

Question 4

Zero moments about two points need not enforce force balance

A planar force system has zero net moment about $A=(0,0)$ and $B=(2,0)\,\mathrm m$. Use $\tau_B=\tau_A-(\vec r_B-\vec r_A)\times\vec F_{net}$.

1. What component of the net force is constrained by these observations?

2. Give a nonzero-force example satisfying both zero-moment conditions.

Question 4 - Solution

Zero moments about two points need not enforce force balance

(a) Derivation and calculation

Use the change-of-origin identity: $$\tau_B=\tau_A-(\vec r_B-\vec r_A)\times\vec F_{\mathrm{net}}.$$ With $\vec r_B-\vec r_A=2\hat x$ and both torques zero, $$0=-2F_y\quad\Longrightarrow\quad\boxed{F_y=0}.$$ There is no constraint on $F_x$.

(b) Result and interpretation

Choose a force $\vec F=5\hat x\ \mathrm N$ at any point $(x,0)$: $$\tau_A=x(0)-0(5)=0,\qquad\tau_B=(x-2)(0)-0(5)=0.$$ Nevertheless, $$\boxed{\vec F_{\mathrm{net}}=5\hat x\ \mathrm N\ne\vec0}.$$ A resultant along the line joining the torque origins is invisible to both moment tests.

Physical check

The two torque equations supply only one independent force-component condition.