Answer to Question #102370 in Mechanics | Relativity for Promise Omiponle

Question #102370
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, ...).
Expert's answer

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}}"

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