Kinetic Energy of a Mass

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Kinetic Energy of a Mass

From Newton's second law, $f=ma=m{\ddot x}$ (introduced in Eq. $\,$ (F.1)), we can derive the formula for the kinetic energy of a mass given its speed $v={\dot x}$ . Let denote a small (infinitesimal) displacement of the mass in the direction. Then we have, using the calculus of differentials,

$\begin{eqnarray*} f(t) &=& m\, {\ddot x}\\ \,\,\Rightarrow\,\,\quad d W\isdef f... ...c{1}{2}{\dot x}^2\right)\\ &=& d\left(\frac{1}{2}m\,v^2\right). \end{eqnarray*}$

Thus, by Newton's second law, a differential of work applied to a mass by force through distance boosts the kinetic energy of the mass by $d(m\,v^2/2)=m\,v\,dv=ma\,dx$ . Therefore, we must have

$\displaystyle E_m(t) = \frac{1}{2}m v^2(t) = \frac{1}{2}{\dot x}^2(t),$

where

denotes the kinetic energy of the mass

traveling at speed

at time

The quantity $dW=f\,dx$ is classically called the virtual work associated with force , and a virtual displacement [560].

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written by 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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Physical Audio Signal Processing
    Elementary Physics, Mechanics, and Acoustics
       Work = Force times Distance = Energy
          Kinetic Energy of a Mass