Physics, 05.02.2020 02:54 yere144k

Two identical, thin rods, each with mass m and length l, are joined at right angles to form an l-shaped object. this object is balanced on top of a sharp edge. if the l-shaped object is deflected slightly, it oscillates.

find the frequency of oscillation.

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The frequency of oscillation of the given L-shaped object made of two rods is $\boxed{\frac{1}{{2\pi }}\sqrt{\frac{{3g}}{{4\sqrt2L}}}}$ .

Further Explanation:

The combination of the two rods perpendicular to each other and making a L-shaped object is as shown in figure attached below.

This L-shaped object rests on a sharp edge and therefore, each rod will make an angle of $45^\circ$ with the sharp edge. When there is a very small deflection in the rod, it will start performing oscillations with a small angle $\theta$ . Then the time period of oscillation of this rod about its center of gravity is given as:

$T=2\pi \sqrt {\frac{I}{{mgd}}}$ …… (1)

Here, is the moment of inertia of the rod, is the distance of the center of gravity from the fixed point and is the time period of oscillation of the L-shaped object.

The moment of inertia of the rod about its one end is given as:

$I=\frac{1}{3}M{L^2}$

For the above mentioned case, the moment of inertia of the two rods about the fixed sharp point is:

$\begin{aligned}I&=\frac{1}{3}m{L^2}+\frac{1}{3}m{L^2}\\&=\frac{2}{3}m{L^2}\\\end{aligned}$

The distance of the center of gravity from the fixed point is:

$d=\frac{L}{{2\sqrt 2 }}$

Substitute the values in equation (1).

$\begin{aligned}T&=2\pi\sqrt{\frac{{\frac{2}{3}m{L^2}}}{{mg\left({\frac{L}{{2\sqrt 2 }}}\right)}}}\\&=2\pi\sqrt{\frac{{4\sqrt 2 L}}{{3g}}}\\\end{aligned}$

The frequency of oscillation of the L-shaped object is:

$\begin{aligned}f&=\frac{1}{T}\\&=\frac{1}{{2\pi}}\sqrt{\frac{{3g}}{{4\sqrt 2 L}}}\\\end{aligned}$

Thus, the frequency of oscillation of the given L-shaped object made of two rods is

$\boxed{\frac{1}{{2\pi }}\sqrt{\frac{{3g}}{{4\sqrt 2 L}}}}$ .

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Answer Details:

Grade: Senior School

Subject: Physics

Chapter: Simple Harmonic Motion

Keywords:

L-shaped object, two identical thin rods, balanced, a sharp edge, deflected slightly, moment of inertia, frequency of oscillation, center of gravity.

Two identical, thin rods, each with mass m and length l, are joined at right angles to form an l-sha

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