2007-11-09 Thin Rods and Density

2007-11-09 Thin Rods and Density - j ; ;..1 :: ~ . I . _0'...

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Unformatted text preview: j ; ;..1 :: ~ . I . _0' :;'J.~ \ .<":. ,.::;.. , .' . .. . ".". .... .'-" n () Thin Rods In many applications. we want to know the center of mass of a rod or a thin strip of metal. In cases like these where we can model the distribution of mass with a continuous function, the summation signs in our formulas become..integrals in a manner we now describe.- Imagine a long. thin strip lying along the x-axis from x = a to x = b and cut into small pieces of mass 6.mt ~y a partition of the interval [a, b]. Xt ~ ~ a ~ t of amt b ( The kth piece is 6.Xt units long and lies approximately Xt units from the origin. Now observe three things. First, the strip's center of mass x is nearly the same as that of the system of point masses we would get by attaching each mass 6.mk to the point Xt: _ system moment x . system mass Second, the moment of each piece of the strip about the origin is approximately Xt6.mt, so the system moment is approxim~tely the sum of the Xt6.lIlt: System moment ~ LXt6.mt....
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