Torque Revisited — Question 1

Question 1

A spatial torque must satisfy two orthogonality checks

A force $\vec F=(4,0,-2)\,\mathrm N$ acts at $\vec r=(2,-1,3)\,\mathrm m$ relative to a fixed origin.

1. Compute the full torque vector.

2. Check that it is perpendicular to both position and force.

Question 1 - Solution

A spatial torque must satisfy two orthogonality checks

(a) Derivation and calculation

Compute the cross product component by component: $$\tau_x=r_yF_z-r_zF_y=(-1)(-2)-3(0)=2,$$ $$\tau_y=r_zF_x-r_xF_z=3(4)-2(-2)=16,$$ $$\tau_z=r_xF_y-r_yF_x=2(0)-(-1)(4)=4.$$ Thus $\boxed{\vec\tau=(2,16,4)\ \mathrm{N\,m}}$ and $\boxed{|\vec\tau|=\sqrt{276}=16.61\ \mathrm{N\,m}}$.

(b) Result and interpretation

Check both perpendicularity conditions: $$\vec\tau\cdot\vec r=2(2)+16(-1)+4(3)=\boxed0,$$ $$\vec\tau\cdot\vec F=2(4)+16(0)+4(-2)=\boxed0.$$ Both vanish, as required for $\vec r\times\vec F$. These checks are independent of the determinant expansion.

Physical check

The torque is perpendicular to the plane containing the position and force vectors.