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Selected Hw Problems for Ch. 9

# Selected Hw Problems for Ch. 9 - 9.6 The Figure shows a...

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9.6 The Figure shows a cubical box that has been constructed from uniform metal plate of negligible thickness. The box is open at the top and has edge length L = 40 cm. Find (a) the x coordinate, (b) the y coordinate, and (c) the z coordinate of the center of mass of the box. A homogeneous symmetric body has its center of mass (com) at its center of symmetry. This means that the plates making up each side have their coms at their respective centers. We can then calculate the com of the entire box as the com due to five point masses (each the mass of a side) at the center point of each plate. Now, formally, com i i i 1 r m r M = G G i i M m = where If the mass of the entire box is designated M then the mass of each side is M/5. The vector equation above is 3 separate eqns, one for each coordinate axis.

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For the x axis, x i xi i 1 1 M M M M M r m r [ (0) (0.2m) (0.2m) (0.2m) (0.4m)] M M 5 5 5 5 5 = = + + + + x 3 1 r [ (0.2m) (0.4m)] 5 5 = + x r 0.20 m = Note that by the symmetry of the box and the location of its center along the x direction (at x = 0.20 m) we could have anticipated this result. We do this for the y direction finding that, y r 0.20 m = For the z axis, z i zi i 1 1 M M M M M r m r [ (0) (0.2m) (0.2m) (0.2m) (0.2m)] M M 5 5 5 5 5 = = + + + +
z 4 r [ (0.2m)] 5 = z 4 r [ (0.2m)] 5 = z r 0.16 m =

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