# Spring (with mass) and Particle Work Energy Theorem Application

I couldn't quite grasp my professor's explanation of this phenomenon. If there is a non-massless spring with a particle attached to it why is there an error in using the work-energy principle if only the particle is considered and no external forces are considered. With that being said, why is it that to properly apply the work-energy principle if the force of the spring is considered to be an external force then the system is no longer a conservative one and the path the spring takes must be considered? My professor was explaining it in words at first which I somewhat understood, but his mathematical explanation that followed was what confused me.

Thank you

• If the spring has mass, does it lose energy as heat when being contracted or absorb heat when it contracts? Yes, which means energy is being exchanged with the environment. It's similar as considering a block moving from point A to B with friction. Some energy also goes into rearranging the lattice in the spring, which is also dependent on time Jul 5, 2017 at 23:03

If it is fixed at one end then not all of the spring moves. Its effective mass $\mu$ is usually about $\frac13$ of its actual mass. The total kinetic energy of the particle-and-spring is $\frac12 (m+\mu) v^2$ where $v$ is the speed of the particle attached to the end of the spring and $m$ is its mass.