Question

A rectangular beam of dimensions b x d is to be cut from a circular log of wood of diameter D for the beam to be strongest in bending, the dimensions will be

• A. $\frac{D}{\sqrt{2}}andD\sqrt{\frac{2}{3}}$
• B. $\frac{D}{\sqrt{3}}andD\sqrt{\frac{2}{3}}$
• C. $\frac{D}{\sqrt{2}}and\sqrt{\frac{2D}{3}}$
• D. $\frac{D}{\sqrt{3}}and\sqrt{\frac{2D}{3}}$

Answer 1

The correct option is B $\frac{D}{\sqrt{3}}andD\sqrt{\frac{2}{3}}$


$\frac{d}{2}=\frac{D}{2}sin\theta$

$\Rightarrow d=Dsin\theta$

$\frac{b}{2}=\frac{D}{2}cos\theta$

$\Rightarrow b=Dcos\theta$

$Z=\frac{b{d}^2}{6}=\frac{D^{3}cos\theta si{n}^2\theta }{6}$

$=\frac{D^3}{6}(cos\theta -co{s}^3\theta )$

For strongest flexural beam,

$\frac{dz}{d\theta }=0$

$\Rightarrow -sin\theta +3co{s}^2\theta sin\theta =0$

$\Rightarrow sin\theta (3co{s}^2\theta -1)=0$

$\Rightarrow cos\theta =\frac{1}{\sqrt{3}}$

$\therefore sin\theta =\sqrt{\frac{2}{3}}$

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

$andd=D\sqrt{\frac{2}{3}}$