Notion - Update docs

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shiffman 2024-02-24 19:18:42 +00:00 committed by GitHub
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@ -905,15 +905,13 @@ function draw() {
<figcaption>Figure 3.20: On the left, the pendulum is drawn rotated so that the arm is the y-axis. The right shows <span data-type="equation">F_g</span> zoomed in and divided into components <span data-type="equation">F_{gx}</span> and <span data-type="equation">F_{gy}</span>.</figcaption>
</figure>
<p>The key here is that the top angle of the right triangle is the same as the angle <span data-type="equation">\theta</span> between the pendulums arm and its resting position. Just as I demonstrated in the discussion of polar coordinates, the sine and cosine functions allow me to separate out the components of the gravity force (the hypotenuse) according to this angle. For <span data-type="equation">F_{gx}</span>, I need to use sine:</p>
<div class="half-width-right">
<figure>
<img src="images/03_oscillation/03_oscillation_18.png" alt="Figure 3.21: F_{gx} is now labeled F_p, the net force in the direction of motion.">
<figcaption>Figure 3.21: <span data-type="equation">F_{gx}</span> is now labeled <span data-type="equation">F_p</span>, the net force in the direction of motion.</figcaption>
</figure>
</div>
<div data-type="equation">\sin(\theta) = F_{gx} / F_g</div>
<p>Solving for <span data-type="equation">F_{gx}</span>, I get this:</p>
<div data-type="equation">F_{gx} = F_g \times \sin(\theta)</div>
<figure>
<img src="images/03_oscillation/03_oscillation_18.png" alt="Figure 3.21: F_{gx} is now labeled F_p, the net force in the direction of motion.">
<figcaption>Figure 3.21: <span data-type="equation">F_{gx}</span> is now labeled <span data-type="equation">F_p</span>, the net force in the direction of motion.</figcaption>
</figure>
<p>Ill now rename this force <span data-type="equation">F_p</span> for <em>force of the pendulum</em>. In Figure 3.21, Ive restored the diagram to its original orientation and relabeled the components. Ive also moved the starting point of <span data-type="equation">F_p</span> from the bottom of the right triangle to the bobs center, to clarify how this force moves the bob.</p>
<p>There it is. The net force of the pendulum that causes the rotation is calculated as follows:</p>
<div data-type="equation">F_p = F_g \times \sin(\theta)</div>

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