PHY183-Lecture22-1 - 1 February 23, 2012 Physics for

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Unformatted text preview: 1 February 23, 2012 Physics for Scientists&Engineers 1 1 Physics for Scientists & Engineers 1 Physics for Scientists & Engineers 1 Spring Semester 2012 Lecture 22 Coordinate Systems and Calculation of Center of Mass Clicker Quiz Clicker Quiz Does a rocket require something to push on in order to move? A. Yes B. No February 23, 2012 Physics for Scientists&Engineers 1 2 February 23, 2012 Physics for Scientists&Engineers 1 3 Volume Integrals Volume Integrals If we want to integrate any function over a volume in three dimensions, we need to find an expression for the volume element dV in an appropriately selected set of coordinates Unless there is an extremely important reason not to, you should always select orthogonal coordinate systems The three sets of orthogonal coordinates in three dimensions that we have introduced so far, Cartesian, cylindrical, and spherical, are also the only ones in practical use February 23, 2012 Physics for Scientists&Engineers 1 4 It is by far easiest to express the volume element dV in Cartesian coordinates Then dV is simply the product of the three individual coordinate elements The three-dimensional volume integral written in Cartesian coordinates becomes Volume Integrals in Cartesian Coordinates Volume Integrals in Cartesian Coordinates f ( r ) dV = f ( r ) dx x min x max y min y max dy dz z min z max V dV = dxdydz February 23, 2012 Physics for Scientists&Engineers 1 5 Volume Integrals in Cylindrical Coordinates Volume Integrals in Cylindrical Coordinates Because we are using the angle as one of the coordinates in cylindrical coordinates, our volume element is not cube-shaped any more For a given differential angle d , the size of the volume element depends on how far away from the z-axis the volume element is located The differential volume is The volume integral is then dV = r dr d dz f ( r ) dV = f ( r ) r dr r min r max min max d dz z min z max V February 23, 2012 Physics for Scientists&Engineers 1 6 Volume Integrals in Spherical Coordinates Volume Integrals in Spherical Coordinates In spherical coordinates we use two angular variables, and The size of the volume element for a given value of the differential coordinates depends on the distance r to the origin as well as the angle relative to the = 0 axis (equivalent to the z-axis in Cartesian or cylindrical coordinates) The differential volume element in spherical coordinates is The volume integral is dV = r 2 dr sin d d f ( r ) dV = f ( r )sin d min max min max d r 2 dr r min r max V 2 February 23, 2012 Physics for Scientists&Engineers 1 7 Position of the Center of Mass Position of the Center of Mass We can then write the position of the center of mass of an...
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PHY183-Lecture22-1 - 1 February 23, 2012 Physics for

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