Torque Revisited — Question 4

Question 4

A central force can have torque about a displaced origin

A particle at $(3,4,0)\,\mathrm m$ feels $\vec F=-2\vec r$ in numerical SI units. Origin $A$ is at $(1,0,0)\,\mathrm m$ relative to $O$.

1. Find torque about $O$ and about $A$.

2. Explain why calling a force central requires naming its center.

Question 4 - Solution

A central force can have torque about a displaced origin

(a) Derivation and calculation

The force is $\vec F=-2\vec r=(-6,-8,0)$ N. About its force center, $$\boxed{\vec\tau_O=\vec r\times(-2\vec r)=\vec0}.$$ Relative to $A$, $\vec r_A=(3,4,0)-(1,0,0)=(2,4,0)$ m. Therefore $$\boxed{\vec\tau_A=(2,4,0)\times(-6,-8,0)=8\hat z\ \mathrm{N\,m}}.$$

(b) Result and interpretation

The origin-shift identity gives the same result: $$\vec\tau_A=\vec\tau_O-\vec a\times\vec F =0-(\hat x)\times(-6\hat x-8\hat y)=8\hat z.$$ The force line passes through $O$, but not through $A$. Calling a force central specifies its center; it does not imply zero torque about every origin.

Physical check

The counterclockwise moment about $A$ follows from the force line passing to its left.