Question

Two uniform boards, tied together with the help of a string, are balanced on a surface as shown in Fig. The coefficient of static friction between boards and surface is 0.5. The minimum value of $\theta$, for which this type of arrangement is possible is :

• A. ${30}^o$
• B. ${45}^o$
• C. ${37}^o$
• D. It is not possible to have this type of balanced arrangement.

Answer 1

The correct option is B ${45}^o$

For the equilibrium of the entire structure, $N_1+N_2=2Mg$

$Mg\left(\frac{1}{2}cos\theta \right)+N_2\times 2x-Mg(\frac{1}{2}cos\theta +2x)=0$

Given $N_2=Mg,N_1=Mg$

For individual boards, $T_1=f_1,N_1=Mg$

and $T_1\times lsin\theta =Mg\frac{1}{2}cos\theta$

$f_1=T_1=\frac{Mg}{2tan\theta }$

For safe equilibrium, $f_1<f_L={\mu }_{s}Mg$

$\frac{Mg}{2tan\theta }={\mu }_{s}Mg\Rightarrow tan\theta >\frac{1}{2{\mu }_s}\Rightarrow \theta ={45}^o$

So the minimum value of $\theta$ for this type of arrangement is ${45}^o$