# Physics Study Guide/Periodic Motion

## Springs

### Hooke's law

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

${\displaystyle k}$ is the spring constant and ${\displaystyle x}$ is the displacement (the force is the restoring force)

For this equation to be approximately accurate, ${\displaystyle x}$ needs to be below the spring's elastic limit. If ${\displaystyle x}$ 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

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

${\displaystyle T}$ is the period, ${\displaystyle m}$ is the mass, and ${\displaystyle k}$ is the spring constant.

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