Question

Two identical rods each of mass m and length L, are joined and then suspended in a vertical plane so as to oscillate freely about an axis normal to the plane of paper passing through 'S' (point of suspension). Find the time period of such small oscillations.

• A. $2\pi$$\sqrt{\frac{17L}{12g}}$
• B. $2\pi$$\sqrt{\frac{13L}{18g}}$
• C. $2\pi$$\sqrt{\frac{17L}{18g}}$
• D. $2\pi$$\sqrt{\frac{17L}{15g}}$

Answer 1

The correct option is C $2\pi$$\sqrt{\frac{17L}{18g}}$

we know that time period of a physical pendulum is given by
$T=2\pi \sqrt{\frac{I_{support}}{mg{l}_{cm}}}{I}_{support}=I_1+I_2=\frac{1}{3}m{L}^2+(\frac{1}{12}m{L}^2+m{L}^2)=\frac{17}{12}m{L}^2{l}_{cm}=\frac{3L}{4}$
$\Rightarrow T=2\pi \sqrt{\frac{\left(\frac{17}{12}m{L}^2\right)}{\left(2m\right)g\left(\frac{3L}{4}\right)}}=2\pi \sqrt{\frac{17L}{18g}}$