6.7 Conservation of energy revisited
Energy is not created or destroyed. The statement
\[
W = E_f – E_i
\]
is an expression of conservation of energy; we account for all energy transfers between kinetic and potential energy within a system, and any energy that is “lost” from the system. Note that that waste energy is not destroyed, it is only removed from the system of interest. The total energy of the universe is still the same.
Example 6.5
A 2 kg block is traveling at a speed of 4 m/s when it hits a spring. The spring compresses, bringing the block to rest in a distance of 12 cm. Throughout this process, 6 J of energy are released as heat. What is the spring constant of the spring?
Start from conservation of energy:
\[
\begin{align*}
W &= E_f – E_i \\
&= U_\textit{sp} – K \\
&= \frac{1}{2}kx^2 – \frac{1}{2}mv^2 \\
\hookrightarrow k &= \frac{2W + mv^2}{x^2} \\
&= \frac{2\left(-6\ \textrm{J}\right) + (2\ \textrm{kg})(4\ \textrm{m/s})^2}{(0.12\ \textrm{m})^2} \\
&= 1.4 \times 10^3\ \textrm{J/m}^2 \\
&= 1.4\ \textrm{kJ/m}^2
\end{align*}
\]
Note that \(W = -6\) J. The negative sign represents energy being dissipated from the system.