Fjäder

Elongationen

\[y = A \sin(\omega t) \]

Hastigheten

\[v = y' \]

\[ v = A \omega \cos(\omega t) \]

\[v_\text{max} = A \omega \]

Accelerationen

\[a = v' = v''\]

\[ a = -A \omega^2 \sin(\omega t) = - \omega^2 y \]

\[a_\text{max}= A\omega^2\]

Kraften

\[ F = -ma \]

\[ F = -m A \omega^2 \sin(\omega t) = -m \omega^2y \]

Vinkelhastigeten, omloppsiden och frekvent

\[\omega^2 = \frac{k}{m} \Rightarrow \omega = \sqrt{\frac{k}{m}}\]

\[ T =\frac{2\pi}{\omega} = 2 \pi \sqrt{\frac{m}{k}} \]

\[ f = \frac{1}{T} = \frac{1}{2\pi \sqrt{\frac{m}{k}}} = \frac{1}{2\pi}\sqrt{\frac{k}{m}}\]

där
\(a =\) accelerationen (m/s)
\(y =\) y led position vid tidpunkten t (m)
\(A =\) amplituden (m)
\( \omega =\) vinkelhastighet (rad/s)
\(t =\) tiden (s)
\(m =\) massa (kg)
\(k =\) fjäderkonstanten (N/m)
\(T =\) omloppstid (s)