Physics Study Guide/Periodic Motion

Springs

Hooke's law

${\vec {F}}=-k{\vec {x}}$

$k$ is the spring constant and $x$ is the displacement (the force is the restoring force)

For this equation to be approximately accurate, $x$ needs to be below the spring's elastic limit. If $x$ is more than the spring's elastic limit, the spring will exhibit "plastic behaviour" meaning that it will not return to its original position.

Potential energy

potential energy stored in a compressed recoiled spring

$U=-\int \limits _{0}^{x_{f}}-k{\vec {x}}\cdot d{\vec {x}}={\tfrac {1}{2}}k{x_{f}}^{2}$

$T$ is the period, $m$ is the mass, and $k$ is the spring constant.

$T=2\pi {\sqrt {\frac {m}{k}}}$