JEE Main · 2012 · 29 Apr 2012 · Offline · Elasticity & SHM

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

1/ e

of original) only in the period between

t =

Os 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:

(A) $\frac{0.693}{b}$
(B)b
(C) $\frac{1}{b}$
(D) $\frac{2}{b}$ ✓ Correct

Answer: (D)

Solution

A bob feeling a viscous drag proportional to its speed swings with an amplitude whose envelope decays as

e^{- b t /2}

, where

b

is the drag constant per unit mass. The amplitude falls to

1/ e

of its starting value when the exponent reaches

- 1

, that is when

e^{- b τ /2} = e^{- 1}

. Equating the exponents gives

\frac{b τ}{2} = 1

, so the average life of the pendulum is

τ = \frac{2}{b}

seconds.

JEE Main · 2012 · 29 Apr 2012 · Offline · Single correct option

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JEE Main · 2012 · 29 Apr 2012 · Offline · Elasticity & SHM

Solution

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