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Energy Conservation in the Mass-Spring System

Recall that Newton's second law applied to a mass-spring system, as in §B.1.4, yields

$\displaystyle f_m(t) + f_k(t) = 0, \quad \forall t, $

which led to the differential equation obeyed by the mass-spring system:

$\displaystyle m{\ddot x}(t) + k\,x(t) = 0 \quad \forall t $

Multiplying through by $ {\dot x}(t)=v(t)$ gives

\begin{eqnarray*} 0 &=& m{\ddot x}(t){\dot x}(t) + k\,x(t){\dot x}(t)\\ &=& m\... ...{d}{dt} \left[ E_m(t) + E_k(t) \right]\\ &=& \frac{d}{dt} E(t). \end{eqnarray*}

Thus, Newton's second law and Hooke's law imply conservation of energy in the mass-spring system of Fig.B.2.

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