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PhysicsEMI/ACDamped SHM and forced oscillationsHard2 minQB

PhysicsHardnumerical

Passage / Comprehension

A uniform rod of mass m and length $ℓ$ is free to rotate about an axis passing through its end. Neglect gravity. There is a spring of constant k at $\frac{ℓ}{2}$ from axis and damper of $F = - cv$ at the free end. If rod is displaced slightly, it executes damped oscillations. Take $\frac{k}{m} = \frac{25}{3} \frac{c ^{2}}{m ^{2}}, \frac{c}{m} = 10/ sec$ .

If the distance of spring from the hinge is $n ℓ$ , the motion is critically damped. Find n .

Options:

Answer:

0.30

Solution:

$\frac{m ℓ ^{2}}{3} \frac{d ^{2} θ}{dt ^{2}} + c ℓ^{2} \frac{d θ}{dt} + kx^{2} θ = 0$ if critically damped, $ω_{0} = r$

$\frac{3 kx ^{2}}{m ℓ ^{2}} = \frac{3 c ℓ ^{2}}{2 m ℓ ^{2}} 3 \times \frac{25}{3} \frac{c ^{2}}{m ^{2}} \frac{x ^{2}}{ℓ ^{2}} = \frac{3 c}{2 m} x = \frac{3}{10} ℓ = 0.3 ℓ$

Stream:JEESubject:PhysicsTopic:EMI/ACSubtopic:Damped SHM and forced oscillations

⏱ 2mℹ️ Source: QB

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