The Requirements of Equilibrium — Question 1

Question 1

A balancing force can leave an uncancelled moment

A planar rigid body receives $\vec F_1=(3,0)\,\mathrm N$ at $(0,0)\,\mathrm m$ and $\vec F_2=(-3,0)\,\mathrm N$ at $(0,1)\,\mathrm m$. No other force or torque acts.

1. Find the force and torque resultants.

2. What pure couple must be added for static equilibrium if the body starts at rest?

Question 1 - Solution

A balancing force can leave an uncancelled moment

(a) Derivation and calculation

The resultant force vanishes: $$\vec F_{\mathrm{net}}=(3,0)+(-3,0)=\boxed{\vec0}.$$ Taking counterclockwise torque as positive, $$\tau_z=\sum(x_iF_{iy}-y_iF_{ix})=0-[1(-3)] =\boxed{+3\ \mathrm{N\,m}}.$$ The system is a nonzero couple, despite force balance.

(b) Result and interpretation

Add a pure clockwise couple $$\boxed{\tau_{\mathrm{add}}=-3\ \mathrm{N\,m}},\qquad\vec F_{\mathrm{add}}=\vec0.$$ Then $$\sum\vec F=\vec0,\quad\sum\tau_z=3-3=0.$$ With initial rest, both translational and rotational static-equilibrium requirements hold.

Physical check

A couple changes the net moment without changing the force resultant.