Schaum's Outline of Finite Mathematics by Seymour Lipschutz

By Seymour Lipschutz

This article covers finite arithmetic. It includes a variety of solved difficulties to assist comprehension.

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Let the masses of the raft and barrel be Mr and Mb, respectively. Ifthe volume of displaced water is initially Vin (a), Archimedes' law requires that the total weight of the raft and barrel equals the weight of the displaced water, whose density is p: (Mr + Mb)g = Vpg. Barrel floats. If the barrel floats, as in (b), with submerged volumes of Vr and Vb for the raft and barrel, respectively, Archimedes' law may be applied to the raft and barrel separately: Barrel: m~ = Vbpg. Raft: m~ = Vrpg, Addition and compare of the two equations shows that: Vr + Vb = V.

A more general case, figure shows a plane vertical surface of arbitrary shape. Note that it is now slightly easier to work in terms of a downward coordinate h. ] Fig. :~~ suriace tiA Total projected : areaA* ----: (a) dA s;::sure\(:,:: Submerged I surface Area ciA of total area A Circled area enlarged (b) Fig. Thrust on surface of uniform cross-sectional shape Thus, the total force is: F = pghc4 = pc4, in which pc is the pressure at the centroid. The advantage of this approach is that the location of the centroid is already known for several geometries.

In 'til qi and mj the direction i indicates the "molecular transport direction", and j indicates the components of the velocity vector for which momentum transport considerations are carried out. The complete equation for 'tij' in the presence of a Newtonian medium can be indicated as follows: k)2 aU- +au;] 2 (au _ - +-po" ~. ~ 3 I) a- 't.. e. force per unit area. It is therefore often designated as "shear stress" and the sign before 11 is chosen positive. This has to be taken into account when comparing representations in this book with corresponding statements in other books.

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