PHY 115
Wave and Oscillation
The solution of the differential equation of SHM:
We know that, the differential equation of simple harmonic motion is,
\(x = A \sin (\omega t + \phi \tag{1}\label{eq:1})\)
♦ The Velocity: \begin{align*} v = & \frac{dx}{dt} = a \cos (\omega t + \phi) \tag{2}\label{eq:2}\\ \implies v = & A \omega \sqrt{1 - \sin^2(\omega t + \phi)} \\ = & \pm A \omega \sqrt{1 - \left(\frac{x}{A}\right)^2} \\ = & \pm A \omega \sqrt{\frac{A^2 - x^2}{A^2}} \\ \therefore v = & \pm \omega \sqrt{A^2 - x^2} \end{align*}
\begin{align*} & v_{min} = 0,~~~ \textrm{when }~ x = a\\ & v_{max} = \omega A , ~~~ \textrm{when} ~ x = 0 \end{align*}
♦ The Acceleration:
\begin{align*}
a = & \frac{dv}{dt} = - a \omega^2 \sin (\omega t + \phi) \\
= & - \omega^2 x ~~\bigg| ~[x = A\sin (\omega t + \phi)] \tag{4}\label{eq:4} \\
\therefore a \propto & - x
\end{align*}
♦ Total Energy: Total Energy = Kinetic energy + Potential energy
Total energy of a body executing SHM:
• Kinetic Energy:
\begin{align*}
v = & \omega \sqrt{A^2 - x^2} = A \omega \cos (\omega t + \phi) \\
K.E = & \frac{1}{2} m v^2 = \frac{1}{2} m \omega^2 (A^2 - x^2) \tag{5}\label{eq:5} \\
K.E = & \frac{1}{2}m v^2 = \frac{1}{2} m A^2 \omega^2 \cos^2 (\omega t + \phi) \tag{6}\label{eq:6}
\end{align*}
• Potential Energy:
\begin{align*}
F = & ma = - m \omega^2 x \\
F = & - m \omega^2 a \sin (\omega t + \phi) \\
& \\
P.E = & - \int_{0}^{x} F~ dx \\
= & - \int_{0}^{x} - m \omega^2 x ~ dx \\
= & \int_{0}^{x} m \omega^2 x ~ dx \\
= & m \omega^2 \int_0^x x ~dx \\
& \frac{1}{2} m \omega^2 x^2 \tag{7}\label{eq:7}
\end{align*}
$$ \therefore P.E = \frac{1}{2} m \omega^2 A^2 \sin^2 ( \omega t + \phi) $$ \begin{align*} \textrm{ Total Energy}~ = & ~\textrm{(5) + (7)} \\ = & \frac{1}{2} m \omega^2 ( A^2 - x^2) + \frac{1}{2} m \omega^2 x^2 \\ = & \frac{1}{2} m \omega^2 A^2 \\ = & \frac{1}{2}k A^2 ~~~~~~~~~~~~~ \bigg| \omega^2 = \frac{k}{m} \\ = & \frac{1}{2} m \left(\frac{2\pi}{T}\right)^2 A^2 \\ = & \frac{1}{2} m (2\pi n )^2 A^2 \\ = & 2 \pi^2 n^2 A^2 m \end{align*}
Assignment: Find out the average value of K.E and P.E of harmonic oscillator.
Post a Comment