ece105: add shm

docs/ce1/ece105.md

@@ -211,3 +211,36 @@ Whether an object *stays* at static equilibrium depends on the
 - It is at **unstable equilibrium** if the object moves away and does not return to equilibrium
 - It is at **stable equilibrium** if the object returns to its original position and equilibrium
 - It is **neutral** if the object does not move
+
+## Simple harmonic motion
+
+!!! definition
+    - The **amplitude** $A$ of a wave is always greater than zero and is equal to the height of the wave above the axis.
+    - The **angular frequency** $\omega$ is the angular velocity, which is dependent only on the restorative force.
+    - The **phase constant** $\phi$ is the offset from equilibrium at $t=0$.
+
+Please see [SL Physics 1#Simple harmonic motion](/g11/sph3u7/#simple-harmonic-motion) for more information.
+
+The position of any periodic motion can be represented as a sine or cosine function (adjust phase as needed).
+
+$$x(t)=A\cos(\omega t+\phi)$$
+
+This means that the velocity function has a phase shift of $\frac{\pi}{2}$ and the acceleration function has a phase shift of $\pi$ along with other changes.
+
+SHM is linked to uniform circular motion:
+
+- $\phi$ is the angle from the standard axis
+- $A$ is the radius
+
+The restorative force can be modelled by substituting in $a(t)$ into $F=ma$
+
+$$F=-m\omega^2x(t)$$
+
+Because restoring force is proportional to the negative position for **smaller displacements**, $F=-Cx(t)$.
+
+Torque is also linear: $\tau=-k\theta$
+
+!!! warning
+    For small angles, $\sin\theta = \theta$.
+
+$$\omega=\sqrt{\frac{C}{m}}$$