By Xin-She Yang

ISBN-10: 1904602568

ISBN-13: 9781904602569

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**Additional info for Applied Engineering Mathematics**

**Sample text**

The interparticle collisions dissipate energy while conserving the total angular momentum. Laplace in 1802 showed that such rings could not be solid because the tensile strength of the known materials was too small to resist tidal forces from Saturn. Later, Niaxwell in 1890 showed that a fluid or gaseous ring was unstable, therefore, the rings must be particulate. Suppose the whole particulate system consists of N particles (i = 1, 2, ... , N). Its total angular momentum is h. By choosing a coordinate system so that (x, y) plane coincides with the plane of the rings, and the z-axis (along k direction) is normal to this plane.

If a, {3 are not all zeros, then these two vectors are linearly dependent. Two linearly dependent vectors are parallel (a//b) to each other. Three linearly dependent vectors a, b, c are in the same plane. 3) i=l which is a real number. 1 Vectors When llxll = 1, then it is a unit vector. 6) where a:, {3 are constants. If() is the angle between two vectors a and b, then the dot product can also be written a· b = llall llbll cos(8), 0 ::; () ::; 7r. , () = 1r /2), then we say that these two vectors are orthogonal.

If we now decompose the velocity of each particle into Vi = (vir, Vie, Viz), the total angular momentum is then N h = k · [L miri x vi] i=l N = L mi(ri i=l X Viz)· k + N N i=l i=l L mi(ri x Vir)· k + L mi(ri x Vie)· k. 65) The first two terms disappear because Viz is parallel to k and axial velocity does not contribute to the angular momentum. 3 Applications Vector Analysis where 1/J(ri) is the potential per unit mass due to Saturn's gravity. The interparticle collisions will dissipate the energy, therefore, the system will evolve towards an energy minimum.

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