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Physical Audio Signal Processing
    Elementary Physics, Mechanics, and Acoustics
       Rigid-Body Dynamics
          Torque

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Torque

Figure B.7: Application of torque $ \tau $ about the origin given by a tangential force $ f_t$ on a lever arm of length $ R$.
\includegraphics[width=1.1in]{eps/torque}

When twisting things, the rotational force we apply about the center is called a torque (or moment, or moment of force). Informally, we think of the torque as the tangential applied force $ f_t$ times the moment arm (length of the lever arm) $ R$

$\displaystyle \tau \isdefs f_tR \protect$ (B.26)

as depicted in Fig.B.7. The moment arm is the distance from the applied force to the point being twisted. For example, in the case of a wrench turning a bolt, $ f_t$ is the force applied at the end of the wrench by one's hand, orthogonal to the wrench, while the moment arm $ R$ is the length of the wrench. Doubling the length of the wrench doubles the torque. This is an example of leverage. When $ R$ is increased, a given twisting angle $ \theta$ is spread out over a larger arc length $ R\theta$, thereby reducing the tangential force $ f_t$ required to assert a given torque $ \tau $.

For more general applied forces $ \underline{f}\in{\bf R}^3$, we may compute the tangential component $ \underline{f}_t$ by projecting $ \underline{f}$ onto the tangent direction. More precisely, the torque $ \tau $ about the origin $ \underline{0}$ applied at a point $ \underline{x}\in{\bf R}^3$ may be defined by

$\displaystyle \underline{\tau}\isdefs \underline{x}\times \underline{f} \protect$ (B.27)

where $ \underline{f}$ is the applied force (at $ \underline{x}$) and $ \times$ denotes the cross product, introduced above in §B.4.12.

Note that the torque vector $ \underline{\tau}$ is orthogonal to both the lever arm and the tangential-force direction. It thus points in the direction of the angular velocity vector (along the axis of rotation).

The torque magnitude is

$\displaystyle \tau \isdefs \vert\vert\,\tau\,\vert\vert \eqsp \vert\vert\,\unde... ...{x}\,\vert\vert \cdot \vert\vert\,\underline{f}\,\vert\vert \cdot\sin(\theta), $

where $ \theta$ denotes the angle from $ \underline{x}$ to $ \underline{f}$. We can interpret $ \vert\vert\,\underline{f}\,\vert\vert \sin(\theta)$ as the length of the projection of $ \underline{f}$ onto the tangent direction (the line orthogonal to $ \underline{x}$ in the direction of the force), so that we can write

$\displaystyle \tau \eqsp \vert\vert\,\underline{x}\,\vert\vert \cdot \vert\vert\,\underline{f}_t\,\vert\vert $

where $ \vert\vert\,\underline{f}_t\,\vert\vert = \vert\vert\,\underline{f}\,\vert\vert \sin(\theta)$, thus getting back to Eq.$ \,$(B.26).

Previous: Rotational Kinetic Energy Revisited
Next: Newton's Second Law for Rotations

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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