Question

A uniform rod $OB$ of length $1m$, cross-sectional area $0.012m^2$ and relative density $2.0$ is free to rotate about $O$ in the vertical plane. The rod is held with a horizontal string $AB$ which can withstand a maximum tension of $45N$. The rod and string system is kept in water as shown in the figure. The maximum value of angle $\alpha$ which the rod can make with vertical without breaking the string is

• A. ${45}^o$
• B. ${37}^o$
• C. ${53}^o$
• D. ${60}^o$

Answer 1

The correct option is B ${37}^o$

R.E.F image

F $=$ upthrust

about $O$,

Anti clock wise torque of $T$ and

upthrust=clock wise torque of

mg.

$T\left(l\cos \alpha \right)=mg\left(\frac{l}{2}\sin \alpha \right)-f\left(\frac{l}{2}\sin \alpha \right)$

$\tan \alpha =\frac{2T}{mg-f}$

$(\tan \alpha {)}_{max}=\frac{2T_{max}}{mg-F}$

$=\frac{2\times 45}{1\times 0.012\times 10}(2000-1000)$

$=0.75\left(or\right)\frac{3}{4}$

${\alpha }_{max}={37}^0$