Rumus Khusus Kecepatan dan Simpangan

Rumus Khusus Kecepatan dan Simpangan

HUBUNGAN KECEPATAN DAN SIMPANGAN

$$v = \omega \:.\: \sqrt{A^2 - y^2}$$

Persamaan simpangan

\begin{equation*} \begin{split} & y = A \:.\: \sin \omega t \\\\ & \sin \omega t = \frac yA \end{split} \end{equation*}

Persamaan kecepatan

\begin{equation*} \begin{split} & v = \frac {dy}{dt} = \omega \:.\: A \:.\: \cos \omega t \\\\ & \cos \omega t = \frac {v}{\omega \:.\: A} \end{split} \end{equation*}

\begin{equation*}
\begin{split}
& \sin^2 \omega t + \cos^2 \omega t = 1 \quad {\color {blue} \text{(Identitas Trigonometri)}} \\\\
& \left( \frac yA \right)^2 + \left( \frac {v}{\omega \:.\: A} \right)^2 = 1 \\\\
& \frac {y^2}{A^2} + \frac {v^2}{\omega^2 \:.\: A^2} = 1 \\\\
& \frac {\omega^2 \:.\: y^2}{\omega^2 \:.\: A^2} + \frac {v^2}{\omega^2 \:.\: A^2} = 1 \\\\
& \frac {\omega^2 \:.\: y^2 + v^2}{\omega^2 \:.\: A^2} = 1 \\\\
& \omega^2 \:.\: y^2 + v^2 = \omega^2 \:.\: A^2 \\\\
& v^2 = \omega^2 \:.\: A^2 - \omega^2 \:.\: y^2 \\\\
& v^2 = \omega^2 \:.\: \left(A^2 - y^2 \right) \\\\
& \bbox[5px, border: 2px solid magenta] {v = \omega \:.\: \sqrt{A^2 - y^2}}
\end{split}
\end{equation*}