Torque — Question 8

Question 8

A torque requirement leaves the radial force undetermined

A force acts at $\vec r=(0.30,0.40)\,\mathrm m$ and must produce $\tau_z=5.0\,\mathrm{N\,m}$. Its in-plane direction is free.

1. Find the force of minimum magnitude meeting the requirement.

2. Describe the family of other forces with exactly the same torque.

Question 8 - Solution

A torque requirement leaves the radial force undetermined

(a) Derivation and calculation

Resolve the force along the radial and positive tangential unit vectors: $$r=\sqrt{0.30^2+0.40^2}=0.50\ \mathrm m,\quad \hat e_r=(0.6,0.8),\quad\hat e_\theta=(-0.8,0.6).$$ $$\tau_z=rF_\theta\Rightarrow F_\theta=\frac{5}{0.50}=10\ \mathrm N, \qquad\boxed{\vec F_{\min}=10\hat e_\theta=(-8,6)\ \mathrm N}.$$

(b) Result and interpretation

The radial component is unrestricted by torque: $$\boxed{\vec F=(-8,6)+c(0.6,0.8)\ \mathrm N},\quad c\in\mathbb R.$$ $$|\vec F|^2=10^2+c^2\ge100,\qquad \vec r\times(c\hat e_r)=\vec0.$$ The minimum occurs at $c=0$ because the components are orthogonal.

Physical check

The proposed minimum gives $0.30(6)-0.40(-8)=5$ N m exactly.