Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Moments of Force

The turning effect of a force about a point, calculated as force multiplied by the perpendicular distance: \vec{M} = \vec{r} \times \vec{F}.
📜

The statement of the theorem

Let R3\mathbb{R}^3 be the Euclidean space equipped with a standard orthonormal basis {i^,j^,k^}\{\hat{i}, \hat{j}, \hat{k}\} and the dot product A⃗⋅B⃗\vec{A} \cdot \vec{B} and cross product A⃗×B⃗\vec{A} \times \vec{B}. Consider a force F⃗\vec{F} applied at a point PP relative to an origin O⃗\vec{O}. The position vector r⃗\vec{r} is defined as r⃗=P⃗−O⃗\vec{r} = \vec{P} - \vec{O}. The moment of force M⃗\vec{M} about the origin O⃗\vec{O} is defined by the vector cross product: M⃗=r⃗×F⃗\vec{M} = \vec{r} \times \vec{F}. In Cartesian coordinates, if r⃗=⟨rx,ry,rz⟩\vec{r} = \langle r_x, r_y, r_z \rangle and F⃗=⟨Fx,Fy,Fz⟩\vec{F} = \langle F_x, F_y, F_z \rangle, the moment vector M⃗=⟨Mx,My,Mz⟩\vec{M} = \langle M_x, M_y, M_z \rangle is given by the determinant expansion: M⃗=∣i^j^k^rxryrzFxFyFz∣\vec{M} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ r_x & r_y & r_z \\ F_x & F_y & F_z \end{vmatrix} M⃗=(ryFz−rzFy)i^+(rzFx−rxFz)j^+(rxFy−ryFx)k^\vec{M} = (r_y F_z - r_z F_y) \hat{i} + (r_z F_x - r_x F_z) \hat{j} + (r_x F_y - r_y F_x) \hat{k}. The magnitude of the moment is M=∣r⃗∣∣F⃗∣sin⁡(θ)M = |\vec{r}| |\vec{F}| \sin(\theta), where θ\theta is the angle between r⃗\vec{r} and F⃗\vec{F}. Alternatively, M=Mx2+My2+Mz2M = \sqrt{M_x^2 + M_y^2 + M_z^2}. This quantity represents the rotational tendency (torque) of F⃗\vec{F} about $\vec{O}.
Source: Wikipedia