CrossBeam  Applet link What: CrossBeam is an applet that calculates geometrical quantities associated with a 2-Dimensional area or combination of areas. The area(s) can be of arbitrary shape, assembled from individual square units. The following geometric quantities are instantly calculated with the addition or subtraction of each area unit: Area - total area of the selected shape, easily calculated as the total number of unit areas. Mnt of Inertia - the area moment of inertia or second moment of area, analogous to the mass moment of inertia traditionally calculated in the dynamics of masses in rotation. The value is calculated with respect to a given axis - in this case the horizontal axis through the center of area (or centroid) of the entire figure. According to the parallel axis rule, each element of area contributes a value calculated about it’s own centroid (1/12 units4 for a unit square) plus the area/distance quantity Ad2 where A is the area of the piece and d is the distance between the local centroid and the chosen axis. Center - coordinates of the center of area (centroid) of the selected shape. A vertical and a horizontal black line run mark these coordinates on the grid. The horizontal line is titled the neutral axis. Coordinates correspond with the central points where unit areas can be placed and run from left (1) to right (20) and top (1) to bottom (20). Note that a value of x.5 corresponds to the dividing line between two grid areas (eg 8.5 would divide units placed at 8 and 9). Y-bar top - the distance from the neutral axis to the topmost edge of the selected shape. Y-bar bottom - the distance from the neutral axis to the bottommost edge of the selected shape. Section Modulus - the area moment of inertia divided by the value of Y-bar top. In addition to the value calculations and the centroidal axes, the applet also graphically displays a footprint of the selected shape (black marks along the bottom and right sides of the grid), identifying all rows and columns containing at least one unit area. How: To add or remove units of area to the grid, the user should position the cursor over or near a grid area (centrally marked when empty by a faint grey circle) and press the mouse button. A blue square of unit area will either appear or disappear. Holding the mouse button down and dragging the mouse will continue to add or remove unit areas. The ‘Hide Data/Show Data’ button will toggle the display of the ‘Information’ box on the right hand side. This can be useful for encouraging calculation of the various quantities by hand. The ‘Flip Beam’ button will reverse the horizontal and vertical coordinates of each beam unit, creating a mirror image of the beam reflected about the diagonal axis from upper left to lower right. Note that the area of the beam will not change, but the other calculated quantities very likely will. The ‘Reset’ button clears all unit areas from the grid and resets the calculated quantities to zero. Why: Under particular conditions, the center of area - demonstrated by the applet - will correspond with the center of volume (given uniform depth in the 3rd dimension) and center of mass (for materials of uniform density) of an actual physical system. The location of the center of mass relative to the supporting base of a figure determines whether the figure will stand naturally under the influence of gravity. The Takes Two to Topple web lesson helps middle school students explore this concept using the applet and Lego brand toy bricks. The strength of structural beams subject to bending loads is dependent upon the cross-sectional profile of the beam and can be directly related to the quantities calculated by the applet. For example, the amount a beam deflects (moves in the direction of the bending load) under given loading conditions is directly proportional to the area moment of inertia. Traditional beam shapes such as the I-beam and box beam balance the opposing design criteria of maximum strength (through high MOI values) and minimum material cost (a direct function of minimized cross-sectional area). Undergraduate engineering students may have interest in a quantitative analysis of these values. A qualitative introduction for middle or high school students is available in the Beam Me Down web lesson.http://www.shodor.org/~grl/Jcodes/CrossBeam/http://en.wikipedia.org/wiki/Second_moment_of_areahttp://www.shodor.org/succeed/curriculum/EIT/pages/toppledir/index.htmlhttp://www.shodor.org/succeed/curriculum/EIT/pages/beamdir/index.htmlshapeimage_2_link_0shapeimage_2_link_1shapeimage_2_link_2shapeimage_2_link_3