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Q. Find the dimension of strongest rectangular beam that can be cut out of log of 25 mm diameter.
Sol.: b2 + d2 = 252
d2 = 252 - b2
As M/I = σ/y; M = σ(I/y) = σ.Z
M is maximum when Z is maximum Z = I/y = (bd3/12)/(d/2) = bd2/6 = b.(252 - b2)/6
The value of Z maximum at dZ/db = 0;
That is, d/db[252b/6 - b3/6] = 0
252/6 - 3b2/6 = 0b2 = 625/3;
b = 14.43 mm .......ANS
d = 20.41 mm .......ANS
vapour compression
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