1. The problem statement, all variables and given/known data
An iron rod of length L is hung at a common point with threads of length 'l' which are attached to the two ends of the rod. The rod is displaced a bit in the plane of the threads. What is the length of the threads if the period of the swinging of the rod is the least, and what is this period?
2. Relevant equations
3. The attempt at a solution
Moment of Inertia of the rod about O is $$ I = \frac{ML^2}{3}+My^2 $$
Write torque equation , ## I\ddotθ = -Mgyθ ##
$$ \ddotθ = -\frac{Mgy}{I}θ $$
$$ ω^2 = \frac{Mgy}{I} $$
$$ ω = \sqrt{\frac{Mgy}{I}} $$
$$ T = 2\pi \sqrt{\frac{I}{Mgy}} $$
$$ T = 2\pi \frac{(l^2+\frac{L^2}{12})^\frac{1}{2}}{g(l^2-\frac{L^2}{4})^\frac{1}{4}} $$
Should I minimize this ? This looks a little more complicated than usual .
An iron rod of length L is hung at a common point with threads of length 'l' which are attached to the two ends of the rod. The rod is displaced a bit in the plane of the threads. What is the length of the threads if the period of the swinging of the rod is the least, and what is this period?
2. Relevant equations
3. The attempt at a solution
Moment of Inertia of the rod about O is $$ I = \frac{ML^2}{3}+My^2 $$
Write torque equation , ## I\ddotθ = -Mgyθ ##
$$ \ddotθ = -\frac{Mgy}{I}θ $$
$$ ω^2 = \frac{Mgy}{I} $$
$$ ω = \sqrt{\frac{Mgy}{I}} $$
$$ T = 2\pi \sqrt{\frac{I}{Mgy}} $$
$$ T = 2\pi \frac{(l^2+\frac{L^2}{12})^\frac{1}{2}}{g(l^2-\frac{L^2}{4})^\frac{1}{4}} $$
Should I minimize this ? This looks a little more complicated than usual .
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