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Damped and Forced Oscillations

If a simple pendulum has significant amplitude (up to a factor

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Hard

Solving time: 5 mins

If a simple pendulum has significant amplitude (up to a factor of $1/ e$ of original) only in the period between $t = 0 s$ to $t = τ s$ , then $τ$ may be called the average life of the pendulum. When the spherical bob of the pendulum suffers a retardation (due to viscous drag) proportional to its velocity, with ' $b$ ' as the constant of proportionality, the average life time of the pendulum is (assuming damping is small) in seconds

0.693/ b

b

1/ b

2/ b

Views: 6,013 students

Updated on: Nov 3, 2023

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Amelia Discussed

If a simple pendulum has significant amplitude (up to a factor of

1/ e

of original) only in the period between

t = 0 s

t = τ s

, then

τ

may be called the average life of the pendulum. When the spherical bob of the pendulum suffers a retardation (due to viscous drag) proportional to its velocity, with '

b

' as the constant of proportionality, the average life time of the pendulum is (assuming damping is small) in seconds

11 mins ago

Discuss this question LIVE

11 mins ago

Text solutionVerified

As retardation

= b v

∴

retarding force

= mb v

∴

net restoring torque when angular displacement is given by
Solution Image

= - m g l sin sin θ + mb v l

∴ I α = - m g l sin sin θ + mb v l

where,

I = m l^{2}

∴ \frac{d ^{2} θ}{d t ^{2}} = α = - \frac{g}{l} sin sin θ + \frac{b v}{l}

for small damping, the solution of the above differential equation will be

∴ θ = θ_{0} e^{- \frac{b r}{2}} sin sin (ω t + ϕ)

∴

angular amplitude will be

= θ . e^{\frac{- b t}{2}}

According to question, in

τ

time (average life-time),
Angular amplitude drops to

\frac{1}{e}

value of its original value

∴ \frac{θ _{0}}{e} = θ_{0} e^{\frac{b τ}{2}} \Rightarrow \frac{b τ}{2} = 1

∴ τ = \frac{2}{b}

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Question Text	If a simple pendulum has significant amplitude (up to a factor of $1/ e$ of original) only in the period between $t = 0 s$ to $t = τ s$ , then $τ$ may be called the average life of the pendulum. When the spherical bob of the pendulum suffers a retardation (due to viscous drag) proportional to its velocity, with ' $b$ ' as the constant of proportionality, the average life time of the pendulum is (assuming damping is small) in seconds
Updated On	Nov 3, 2023
Topic	Oscillations
Subject	Physics
Class	Class 11
Answer Type	Text solution:1 Video solution: 3
Upvotes	349
Avg. Video Duration	5 min

Text solutionVerified

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