Question

Two identical thin rods of length L (and equal mass) are joined at right angles to form an L-shaped object. This object is balanced in a vertical plane on top of a sharp edge. The L-shaped object oscillates as a physical pendulum when it is slightly deflected (small angle). Starting with T=2pi(I/mgd)^(1/2), derive an expression for the frequency of the oscillation. Write the result in terms of thequantities given in the problem and, perhaps, constants (e.g., π, ⅓, g, ...).

Answer 1

The mass of the rod = m

Length of the one rod = L

As per the question, if it is joining at $90^\circ$

Then, Moment of the inertia of the combined system = $\dfrac{mL^2 }{3}+\dfrac{mL^2}{3}=\dfrac{2mL^2}{3}$

The effective length of the pendulum =$L\cos45^\circ=\dfrac{L}{\sqrt{2}}$

Now, we know that time period $T=2\pi\sqrt{\dfrac{I}{mgl}}$

$T=2\pi\sqrt{\dfrac{2mL^2\sqrt{2}}{3\times 2mgL}}$

$\Rightarrow T=2\pi\sqrt{\dfrac{L\sqrt{2}}{3\times g}}$

$\Rightarrow T=2 \pi\sqrt{\dfrac{\sqrt{2}L}{3g}}$