Equations of Motion: Harmonic Oscillator

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Section 11.1 Equations of Motion: Harmonic Oscillator

Using conservation of energy, let’s explore the motion of a classical harmonic oscillator.

The statement of energy conservation:

\begin{equation} E = T + U\tag{11.1.1} \end{equation}

becomes

\begin{equation} E = \frac12 m \dot{x}^2 + \frac12 (x-x_0)^2\tag{11.1.2} \end{equation}

(11.1.2) can be solved for $\dot{x}$ to give:

\begin{equation} \dot{x} = \pm\sqrt{\frac{2}{m}\bigl(E-\frac12 (x-x_0)^2\bigr)}\tag{11.1.3} \end{equation}

Activity 6.

Explore how the kinetic energy $\frac12 m \dot{x}^2$ of a harmonic oscillator is affected by the shape of the potential (shown in blue) (which depends on the parameters $k$ and $x_0$) and the value of the total energy (shown in green).

Figure 13. The potential for the classical harmonic oscillator is shown in blue. For a given potential, the total energy $E\text{,}$ shown in green will determine the resulting motion.

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