>[!summary] SHM is the ideal type of motion > The type of wave this would create is this: $x(t) = Acos(\omega t + \phi)$ > For a simple pendulum: $\omega = \sqrt{\frac{g}{L}}$ > For a real pendulum: $\omega = \sqrt{\frac{mgd}{I}}$ # Deriving SHM equation Simple harmonic motion is the motion an ideal spring makes on a frictionless floor. We assume this spring attached to some mass always reflects and follows hooks law as described by the figure below. ![[SHM_1.png|400]] [^1] >[!note] Explanation >First image: >SHM for a spring at rest. Notice its a equilibrium L distance from the wall > Second image: SHM as the motion reaches its max stretch and starts to reflect > Third image: SHM as the motion reaches it max compression and starts to reflect We often imagine this effect on a unit circle where the position can be defined at point P. ![[SHM_2.png]] [^2] Note than our angle will be defined as $\theta = \omega t + \phi$ If we want to define the position (P) at any point in time notice how at t = 0 our position in time is defined (x) so we get a cos graph. $\begin{array}{c} x(t) = Acos(\theta) \\ x(t) = Acos(\omega t + \phi) \end{array}$ ## Energy Of A SHM Our kinetic energy and potential energy can be rewritten from the equation above derived earlier. $\begin{array}{c} K = \frac{1}{2}m v^2 \quad \text{Can be rewritten as} \\ K = \frac{1}{2}m \ddot{x} \\ x(t) = Acos(wt+\phi) \\ \dot{x} = -Asin(\omega t + \phi) \\ \\ K = \frac{1}{2}m(Asin(\omega t + \phi))^2 \\ K = \frac{1}{2}mAsin^2(\omega t + \phi) \end{array}$ The same can be done for potential energy but wont be derived in this note # Simple Pendulum A simple pendulum is a block with a **mass** on a **massless** string. For a simple pendulum to be true we only imagine it over small amplitudes only considering the x direction. ![[SHM_3.png]] [^3] >[!note] Explanation A mass swinging over a length on a massless string. Assume small angle approx. Consider the expression where we only consider the x direction. $\begin{array}{c} F = ma \\ -mgsin(\theta) = ma \\ \text{Imagine sin ove small angles as just $\theta$} \\ -mgsin(\theta) = m\ddot{x} \\ -mg\theta = \frac{\mathrm{d^2 s} }{\mathrm{d}t ^2} \\ \text{Remember that because this creats an arc $s = L\theta$} \\ mL\frac{\mathrm{d^2 \theta} }{\mathrm{d}t ^2} = -mg\theta \\ \frac{\mathrm{d^2 \theta} }{\mathrm{d}t ^2} = - \frac{g}{L}\theta \end{array}$ So for any value $\omega$: $\omega = \sqrt{\frac{g}{L}}$ # Physical Pendulum A physical pendulum is any real pendulum where it has an extended body, like a string and mass with some mass. ![[SHM_4.png]] [^3] Again deriving this expression well assume for small values of angles and only consider the x-direction of motion. $\begin{array}{c} \tau = I\alpha \\ -mgdsin(\theta) = I \frac{\mathrm{d^2 \theta} }{\mathrm{d}t ^2} \\ \text{Well assume for small values of angle} \\ -mgd\theta = I \frac{\mathrm{d^2 \theta} }{\mathrm{d}t ^2} \\ \frac{-mgd}{I}\theta = \frac{\mathrm{d^2 \theta} }{\mathrm{d}t ^2} \end{array}$ so $\omega = \sqrt{\frac{mgd}{I}}$ [^1]: Taken from https://tikz.net/dynamics_spring/ by Izaak Neutelings (September 2020) [^2]: Taken from https://tikz.net/dynamics_oscillator/ by Izaak Neutelings (October 2020) [^3]: Taken from https://tikz.net/dynamics_pendulum/ by Izaak Neutelings (October 2020) --- > 🧠 Enjoy this walkthrough? [Support Math & Matter](https://github.com/rajeevphysics/Obsidan-MathMatter) with a star and help others learn more easily. ---