Torque — Question 3

Question 3

A couple has zero resultant force and a nonzero moment

Two forces act in the plane: $(0,8)\,\mathrm N$ at $(0.50,0)\,\mathrm m$ and $(0,-8)\,\mathrm N$ at $(-0.50,0)\,\mathrm m$.

1. Find the resultant force and total torque about the origin.

2. Find total torque about a pivot at $(2,1)\,\mathrm m$ and explain its invariance.

Question 3 - Solution

A couple has zero resultant force and a nonzero moment

(a) Derivation and calculation

Sum both the forces and the moments: $$\boxed{\vec F_{\mathrm{net}}=(0,8)+(0,-8)=\vec0}.$$ $$\boxed{\tau_{O,z}=0.50(8)+(-0.50)(-8)=8.0\ \mathrm{N\,m}}.$$ The two forces form a couple with magnitude $Fd=8(1)=8$ N m.

(b) Result and interpretation

Relative to the new pivot, the position vectors are $(-1.5,-1)$ and $(-2.5,-1)$ m: $$\boxed{\tau_{A,z}=(-1.5)(8)+(-2.5)(-8)=-12+20=8.0\ \mathrm{N\,m}}.$$ In general, $$\vec\tau_A=\vec\tau_O-\vec a\times\vec F_{\mathrm{net}}=\vec\tau_O.$$ A zero resultant makes the couple independent of the chosen origin.

Physical check

Equal and opposite forces cancel translation but their rotational effects reinforce.