The Requirements of Equilibrium — Question 9

Question 9

Planar balance tests can miss an out-of-plane couple

A spatial rigid body has forces $+2\hat z\,\mathrm N$ at $(0,1,0)\,\mathrm m$ and $-2\hat z\,\mathrm N$ at $(0,-1,0)\,\mathrm m$. No other loads act.

1. Compute the full force and torque resultants.

2. Would checking only $F_x=F_y=\tau_z=0$ diagnose the failure of equilibrium?

Question 9 - Solution

Planar balance tests can miss an out-of-plane couple

(a) Derivation and calculation

The forces cancel: $$\boxed{\vec F_{\mathrm{net}}=2\hat z-2\hat z=\vec0}.$$ Their moments add: $$\vec\tau_1=(1\hat y)\times(2\hat z)=2\hat x,$$ $$\vec\tau_2=(-1\hat y)\times(-2\hat z)=2\hat x,$$ $$\boxed{\vec\tau_{\mathrm{net}}=4\hat x\ \mathrm{N\,m}}.$$

(b) Result and interpretation

The proposed planar tests give $$F_x=F_y=\tau_z=0,$$ so all pass while the spatial body is not in equilibrium. General rigid-body balance requires $$F_x=F_y=F_z=0,\qquad\tau_x=\tau_y=\tau_z=0.$$ The missing $\tau_x$ check detects the nonzero couple.

Physical check

The torque direction follows $\hat y\times\hat z=\hat x$ for both force applications.