By Биленский С.М.
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Prove that ∠ F1 P M = ∠ F2 P N . Solution. How might we prove this property? A brute force solution is to (i) write the equation of an ellipse, (ii) compute the two angles in question, and (iii) verify that the expressions are equal. 2. (a) The optical property; (b) a mechanical proof. approach can lead to a finger-breaking calculation and, to add insult to injury, little understanding would be gained. 2(b)), letting a weighted pulley roll on the string as shown. If we move the pulley left or right, while keeping the string taut, the pulley will trace out an ellipse.
The arc Am B is the least-time path; the light from A does not focus on this arc. By contrast, the arc An B is of extremal, but not shortest time; this property goes together with the existence of a focusing point on this arc. Observing Fermat’s principle in action. 20. 16 This explains why the concave lens defocuses, while the convex lens focuses. 21 shows a ray from the bottom of a pool to the eye. The path shown in the figure is quicker than the straight path, since it “pays” to shorten the “expensive” underwater part where light is slower.
Then the point of tangency of each face with K is that face’s centroid. Proof. We imagine four planes enclosing K and bounding a pyramid-shaped bubble of vacuum. The planes can pass through each other without interaction, but they cannot penetrate K . The 40 CHAPTER 3 air pressure outside the bubble of vacuum forces the planes to press against K . 8 Hence if the pyramid has minimal volume, then the potential energy will be minimal, and hence all planes will be in equilibrium. Thus the outward point pressure on each face at the tangency point balances the uniformly distributed inward pressure.