Mass Kinetic Energy from Virtual Work
From Newton's second law,
(introduced in Eq.
(B.1)),
we can use d'Alembert's idea of virtual work to derive the
formula for the kinetic energy of a mass given its speed
.
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}(t)\\
\,\,\Rightarrow\,\,\quad d W\isde...
...{1}{2}{\dot x}^2\right)\\
&=& d\left(\frac{1}{2}m\,v^2\right).
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img2660.png)
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
. The kinetic energy of a mass moving at
speed
is then given by the integral of all such differential
boosts from 0 to
:
![$\displaystyle E_m(v) = \int_0^v dW = \int_0^v d\left(\frac{1}{2}m \nu^2\right)
= \frac{1}{2}m v^2 = \frac{1}{2}m\,{\dot x}^2,
$](http://www.dsprelated.com/josimages_new/pasp/img2663.png)
![$ E_m(v)$](http://www.dsprelated.com/josimages_new/pasp/img2664.png)
![$ m$](http://www.dsprelated.com/josimages_new/pasp/img6.png)
![$ v$](http://www.dsprelated.com/josimages_new/pasp/img345.png)
The quantity is classically called the virtual work
associated with force
, and
a virtual displacement
[544].
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Energy in the Mass-Spring Oscillator
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Kinetic Energy of a Mass