Consider a pendulum with a bob of mass $m$ attached to a rigid but massless rod with length $L$. Which equation describes the motion of the bob with respect to the vertical? 1. $m\ddot{\theta} = +g\sin\theta$ 2. $m\ddot{\theta} = -g\sin\theta$ 3. $mL\ddot{\theta} = -mg\sin\theta$ 4. $mL\ddot{\theta} = +mg\sin\theta$ 5. Something else Note: Correct answer: C
Let's take the easy route for the moment. <img src="./images/simplify.jpg" align="center" style="width: 400px";/> $\ddot{\theta} \approx -\dfrac{g}{L} \theta$
<img src="./images/morpheus.jpg" align="left" style="width: 400px";/> What is the general solution to: $\ddot{\theta} \approx -\omega^2 \theta$? 1. $\theta(t) = A \cos \omega t$ 2. $\theta(t) = B \sin \omega t$ 3. $\theta(t) = A \cos \omega t + B \sin \omega t$ 4. $\theta(t) = A \cos (\omega t + \delta)$ 5. More than one of these
OMGBBQPIZZA <img src="./images/SHO_everywhere.jpeg" align="center" style="width: 600px";/>
Nature tends to minimize energy <img src="./images/SHO.jpg" align="center" style="width: 400px";/>
Have you worked with phase space before? 1. Yes, and I recall how that works 2. Yes, I think so...ok, actually, maybe... 3. I have no idea what you are talking about, hoss
Now that we have sketched $\langle \dot{x}, \dot{v} \rangle = \langle v,0\rangle$... Sketch $\langle \dot{x}, \dot{v} \rangle = \langle 0,-x\rangle$ in phase space.
What about $\ddot{x} = -\sin{x}$? <img src="./images/theory.png" align="center" style="width: 350px";/>