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 {\vec {\mathrm {d} x}}={\frac {1}{2}}kx_{f}^{2}

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

T=2\pi {\sqrt {m \over k}}

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