diff --git a/docs/ce1/ece105.md b/docs/ce1/ece105.md index cd85880..c9f8cd0 100644 --- a/docs/ce1/ece105.md +++ b/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, $\sin\theta = \theta$. + +$$\omega=\sqrt{\frac{C}{m}}$$