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Kinetic Energy of a Mass

Kinetic energy is energy associated with motion. For example, when a spring uncompresses and accelerates a mass, as in the configuration of Fig.B.2, work is performed on the mass by the spring, and we say that the potential energy of the spring is converted to kinetic energy of the mass.

Suppose in Fig.B.2 we have an initial spring compression by $ A$ meters at time $ t=0$ , and the mass velocity is zero at $ t=0$ . Then from the equation of motion Eq.$ \,$ (B.5), we can calculate when the spring returns to rest ($ x(t)=0$ ). This first happens at the first zero of $ \cos(\omega_0t)$ , which is time $ t=(\pi/2)/\omega_0=(\pi/2)\sqrt{m/k}$ . At this time, the velocity, given by the time-derivative of Eq.$ \,$ (B.5),

$\displaystyle v(t) = -A\omega_0\sin(\omega_0 t),
$

can be evaluated at $ t=(\pi/2)/\omega_0$ to yield the mass velocity $ v[(\pi/2)/\omega_0] = -A\omega_0 = -A\sqrt{k/m}$ , which is when all potential energy from the spring has been converted to kinetic energy in the mass. The square of this value is

$\displaystyle v^2_{\mbox{max}} = A^2\omega_0^2 = A^2\frac{k}{m},
$

and we see that if we multiply $ v^2_{\mbox{max}}$ by $ m/2$ , we get

$\displaystyle \frac{1}{2}m\,v^2_{\mbox{max}} = \frac{1}{2}k\,A^2,
$

which is the initial potential energy stored in the spring. We require this result. Therefore, the kinetic energy of a mass must be given by

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

in order that the kinetic energy of the mass when spring compression is zero equals the original potential energy in the spring when the kinetic energy of the mass was zero. In the next section we derive this result in a more general way.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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