Plane and Solid Analytic Geometry by William F. Osgood

By William F. Osgood

The article of an basic university direction in Analytic Geometry is twofold: it truly is to acquaint the coed with new and engaging and demanding geometrical fabric, and to supply him with strong instruments for the examine, not just of geometry and natural arithmetic, yet in no much less degree of physics within the broadest experience of the time period, together with engineering. to realize this item, the geometrical fabric may be offered within the easiest and such a lot concrete shape, with emphasis at the geometrical content material, and illustrated, every time attainable, via its relation to physics. This precept has been saw during the booklet. therefore, in treating the ellipse, the tools truly utilized in the drafting room for drawing an ellipse from the knowledge as a rule met in descriptive geometry are given a number one position. the theory that the tangent makes equivalent angles with the focal radii is proved routinely: a rope which passes via a pulley has its ends tied on the foci and is drawn taut via a line mounted to the pulley. furthermore, the that means of foci in optics and acoustics is cleariy set forth. A achieve, there's a bankruptcy at the deformations of an elastic airplane less than tension, with symptoms as to the third-dimensional case (pure pressure, etc.). The tools of analytic geometry, even of their least difficult types, make serious calls for at the scholars skill to appreciate the reasoning of upper arithmetic. accordingly, in proposing them for the 1st time, in basic terms algebraic problems, equivalent to are attributable to literal coefficients and lengthy formal computations, may be avoided.
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Sample text

Let us calculate the change in potential energy. -,--- ..... , F' ___ F W' - W = Ph, where P is the weight of the solid and h the increase in the height of its center of gravity. Clearly, h = AA' = BB'. Fig. 12 • , . 32 The Center of Gravity, Potential Encray, and Work Furthermore, supposing the solid to be homogeneous, we can write p = Vi'. <--~ where V is tile volume of the solid and i' its specific weight. vyh. 23) ~d, 'W' =- W""BDF' +. tA,,,,', and, therefote. 24) t of , , that is, W' - W is equal to the differetce the potential energy of the bodies BB' D' D and AA.

S) is the moment of area of the rectangle with respect to the axis of rotation. ~ Let us now substitute an arbitrary figure Q for this rectangle (fig. 24). ". 2 4 z ,. h C He P- H, R2 Fig. 23 approximate each of these strips by the rectangle inscribed in each strip. If n denotes the number of strips, and we allow this number without. bound, the approximations become successively better. We then have " v = lim (,Vl + Vll + ... + Vn) , t' " .... 61), • 51 J • ,The Center of. ~vity. Potential Energy.

S = ,21T' DC· sin fJ· 2Ra . 21), however, tells US that DC s= = R(sin a/a). Consequently, 21TR sin ex sin,8. 2Ra = 21TR· 2R sin a sin,8 . a . Since , " 2R sin a = I, we now have Again referring to the sketch, we note that the 'second factor of this product is. equal tp the altitude of the spherical strip (that is, the projectron of the chdtd"A B onto the diameter PQ). Denoting this altitude by H, we finally' obtain' the formula ' , " ... , --. 3. 30 rotates about the axis 00'. The surface area ,of the resulting solid'is equal to , " ' ( + a)2 ·4 ay'(2) ~ '= (my'(2~ .

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