Question

The width of the strongest beam of rectangular section that can be cut out of a cylindrical log of wood whose diameter is 300 mm would be

• A. 104.5 mm
• B. 156.2 mm
• C. 202.6 mm
• D. 173.2 mm

Answer 1

The correct option is D 173.2 mm

Strongest beam is the one for which section modulus Z is maximum.

Let rectangular section is of width b and depth d,

Diameter of cylindrical log, D = 300 m

$\therefore b^2+d^2=D^2$

$\therefore Z=\frac{b{d}^2}{6}=\frac{b(D^2-b^2)}{6}=\frac{b{D}^2-b^3}{6}$

For maximum Z;

$\frac{\partial Z}{\partial b}=0$

$\Rightarrow \frac{D^2-3b^2}{6}=0$

$\Rightarrow b=\frac{D}{\sqrt{3}}$

$\Rightarrow b=\frac{300}{\sqrt{3}}=173.2mm$