The mathematical mechanic

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[...]... version of the Pythagorean theorem (p 19) The proof of the Pythagorean theorem, described in section 2.2, suggested a kinematic proof of the Pythagorean theorem, described in section 2.6 The motion-based approach makes some other topics very transparent, including • The fundamental theorem of calculus • The computational formula for the determinant • The expansion of the determinant in a row All these are... summarize: the area theorem (2.4) amounts to saying that the pressurized container of the shape shown in figure 2.10 provides zero thrust! A simple physical observation gives a neat mathematical theorem A mathematical “cleanup.” A skeptic may complain about the lack of mathematical rigor in getting to (2.5) Indeed, we had appealed to the law of conservation of energy, which had not been given a precise mathematical. .. follows The law states: the buoyancy force acting on a submerged body (say a rock) equals the weight of the water displaced by the body Proof Imagine replacing the submerged rock with the identically shaped blob of water This blob of water will hover in equilibrium, as mentioned above The buoyancy on the water blob therefore equals the blob’s weight But the rock “feels” the same buoyancy since it has the. .. as the segment executes two simultaneous motions: sliding (in the direction of the segment) and rotating around the trailing point T of the segment The key observation is this: the sliding motion does not affect the rate at which the segment sweeps the area In other words, by subtracting the sliding velocity, and thus making the segment rotate in place around its trailing point, we do not affect the. .. 15 THE PYTHAGOREAN THEOREM a –bc b c, d c ad de t March 25, 2009 d a, b Figure 2.6 The segment moves while remaining parallel to itself The area swept does not depend on the path of the segment 2.5 The Determinant by Sweeping b The determinant a d is, by defnition, the area of the parallelogram c generated by the vectors a, b and c, d This definition leads to the computational formula,4 giving the. .. thus ad −bc It remains to observe that the area swept does not depend on the path of the moving vector a, b , as long as it moves parallel to itself Indeed, the rate of change of the area swept equals the length of the segment times the speed in the perpendicular direction Thus the 4 Some unfortunates, including the author, have been taught the latter formula as the definition but not its geometrical... the parallelogram formed by the last two row vectors in the direction of the x axis by a11 , then in the y direction by a12 , and finally in the z direction by a13 Compare the volume swept with the volume swept by under the “diagonal” translation by a11 , a12 a13 2.6 The Pythagorean Theorem by Rotation Figure 2.7 shows a right triangle executing one full turn around an endpoint of its hypotenuse The. .. hypotenuse and the leg adjacent to the pivot sweep out disks, while the remaining leg sweeps out a ring March 25, 2009 Time: 04:45pm chap02.tex THE PYTHAGOREAN THEOREM We have 17 πa 2 + (area of the ring) = πc2 Proving the Pythagorean theorem amounts to showing that the area of the ring is πb2 How do we prove this directly, without appealing to the theorem? Here is a heuristic argument The ring is... more physical proofs of the Pythagorean theorem are given here, one using springs, and the other using kinetic energy The unifying theme of this chapter is the Pythagorean theorem, although we do go off on a few short tangents 2.2 The “Fish Tank” Proof of the Pythagorean Theorem Let us build a prism-shaped “fish tank” with our right triangle as the base (figure 2.1) We mount the tank so that it can rotate... that the torque of the force around a pivot point P is simply the force’s magnitude times the distance from the line of force to the pivot point The torque measures the intensity with which the force tries to rotate the object it’s applied to around P For convenience, let us assume the force of pressure to be 1 pound per unit length of the wall—we can always achieve it by adjusting water depth The three . observe how the difficulty changes shape in passing from one approach to the other. In the mathematical solution, the work goes into a formal manipulation. In the physical approach, the work goes. 25, 2009 Time: 04:39pm fm.tex THE MATHEMATICAL MECHANIC i March 25, 2009 Time: 04:39pm fm.tex ii March 25, 2009 Time: 04:39pm fm.tex MARK LEVI THE MATHEMATICAL MECHANIC using physical reasoning to. makes some other topics very transparent, including • The fundamental theorem of calculus. • The computational formula for the determinant. • The expansion of the determinant in a row. All these are