By B. Yu. Kogan
Rear conceal notes: "This e-book is an exposition of geometry from the perspective of mechanics. B. Yu. Kogan starts via defining thoughts of mechanics after which proceeds to derive many subtle geometric theorems from them. within the ultimate part, the innovations of strength strength and the guts of gravity of a determine are used to boost formulation for the volumes of solids. those may well previously be derived in basic terms by means of the calculus."
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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~ .