Torque Revisited — Question 3

Question 3

Some specified torque vectors cannot come from a given point force

A force may be chosen freely but must act at $\vec r=(1,2,0)\,\mathrm m$. A designer asks for torque $(2,0,3)\,\mathrm{N\,m}$.

1. Test whether the request is possible.

2. For the alternative target $(2,-1,3)\,\mathrm{N\,m}$, find the smallest-magnitude force.

Question 3 - Solution

Some specified torque vectors cannot come from a given point force

(a) Derivation and calculation

A necessary condition for a single applied force is $$\vec\tau\cdot\vec r=(\vec r\times\vec F)\cdot\vec r=0.$$ For the requested first target, $$(2,0,3)\cdot(1,2,0)=2\ne0.$$ Therefore $\boxed{\text{the first target is impossible}}$.

(b) Result and interpretation

The alternative target satisfies $(2,-1,3)\cdot(1,2,0)=0$. Its transverse force is $$\vec F_\perp=\frac{\vec\tau\times\vec r}{r^2} =\boxed{(-6/5,3/5,1)\ \mathrm N}.$$ Every solution is $\vec F=\vec F_\perp+\lambda\vec r$, and $$F^2=F_\perp^2+\lambda^2r^2\ge F_\perp^2 \Rightarrow\boxed{F_{\min}=\sqrt{14/5}=1.673\ \mathrm N}.$$

Physical check

Direct substitution gives $(1,2,0)\times(-6/5,3/5,1)=(2,-1,3)$.