Torque (source code)
= Torque
{title2=$\boldsymbol\tau_O=\mathbf r\times\mathbf F$}
{wiki}
= Moment of force
{synonym}
<Torque> measures a <force>'s turning effect about a specified origin. The geometry in the <angular momentum> diagram applies with <force> replacing <momentum>: transverse <forces> give positive or negative <torque>, while a radial <force> gives zero. A longer perpendicular <lever arm> makes the same <force> more effective, explaining why pushing near the end of a handle is useful.
For a <force> $\mathbf F$ applied at <position> $\mathbf r$ from a fixed origin $O$ in an <inertial frame>, its <torque>, or <moment of force>, is
$$
\boldsymbol\tau_O=\mathbf r\times\mathbf F,\qquad
|\boldsymbol\tau_O|=rF\sin\phi=F\ell.
$$
Here $F=|\mathbf F|$, $\phi$ is the <angle> between <position> and <force>, and $\ell=r\sin\phi$ is the perpendicular <lever arm>. Direction follows the <right-hand rule>; in the $xy$ plane $\tau_z=xF_y-yF_x$, positive counterclockwise as viewed from $+z$. The <International System of Units> unit is $\mathrm{N\,m}=\mathrm{kg\,m^2/s^2}$. Although <torque> and <energy> have the same <physical dimensions>, <torque> is an <axial vector> and <energy> is a <scalar>; their roles differ.
For $\mathbf r=(2,0,0)\,\mathrm m$ and $\mathbf F=(0,3,0)\,\mathrm N$, $\tau_z=+6\,\mathrm{N\,m}$. Applying the same <force> at $(1,0,0)\,\mathrm m$ gives $+3\,\mathrm{N\,m}$; reversing it gives a negative <torque>; directing it along $\mathbf r$ gives zero. More generally, sum $\mathbf r_i\times\mathbf F_i$ to find the total <torque> about one common origin.
For a constant-<mass> <point mass>, the <product rule> and <Newton's second law> show why net <torque> changes <angular momentum>:
$$
\frac{d\mathbf L_O}{dt}
=\mathbf v\times(m\mathbf v)+\mathbf r\times\frac{d\mathbf p}{dt}
=\mathbf r\times\mathbf F_{\mathrm{net}}
=\boldsymbol\tau_O^{\mathrm{net}}.
$$
The first <cross product> vanishes because its factors are parallel. For a collection with canceling internal <torques>, the remaining sum is external <torque>, giving <conservation of angular momentum> when it vanishes.<University Physics Volume 1 by William Moebs and others/11.2 Angular Momentum>[section 11.2: torque and the angular momentum balance]{ref}
A nonzero <torque> can turn the direction of <angular momentum> as well as change its magnitude. Indeed, the <dot product> gives $d|\mathbf L_O|^2/dt=2\mathbf L_O\cdot\boldsymbol\tau_O$: a perpendicular <torque> changes direction without instantaneously changing the magnitude. This is the geometry behind <gyroscope> <precession>. For a rapidly spinning <symmetric top> with a fixed tip, <centre of mass> distance $d$ from the tip, <mass> $M$, and spin-dominated <angular momentum> magnitude $L\simeq I\omega$, the slow steady <precession> approximation gives $\Omega\simeq Mgd/L$, where $g$ is <gravitational acceleration>. <University Physics Volume 1 by William Moebs and others/11.4 Precession of a Gyroscope>[OpenStax University Physics Volume 1, section 11.4: Precession of a Gyroscope]{ref} This is an approximation with $\Omega\ll\omega$; the <uniform precession of a heavy symmetric top> retains contributions omitted by the spin-dominated estimate.
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