Wood Beam
The strength of a wooden rectangular beam varies jointly as the width and the square of its depth. Which rectangular beam that can be cut from a circular log of radius 10 inches will have maximum strength?
Labels: funphysics
posted by Ragknot at
4:51 AM
5 Comments:
the one with steel reinforcement
yess!!! first one
width = 11.547 inches
breadth = 16.329 inches
"breadth" is another word for width.
Width = sqrt(400/3)
Depth = sqrt(800/3)
The strength of the beam is given:
Str = w*d²
we also have the formula for the radius of the log
10² = (w/2)²+(d/2)²
400 = w²+d²
d² = 400-w²
combining the two formulas we get
Str = 400w-w³
to find the maximum we take the derivative
Str = 400w-w³
Str' = 400-3w²
solve for w when the Str'=0
0 = 400-3w²
w = sqrt(400/3)
substitute it back in to the radius formula to solve for d
400 = w²+d²
400 = (400/3)+d²
d = sqrt(800/3)
w = sqrt(400/3)
what?
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