If a simple pendulum has significant amplitude (up to a factor of 1/e of original) only in the period between t = 0s 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 averatge life time of the pendulum is (assuming damping is small)

Question

If a simple pendulum has significant amplitude (up to a factor of 1/e of original) only in the period between t = 0s 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 averatge life time of the pendulum is (assuming damping is small) (A) (0.693/b) (B) b (C) (1/b) (D) (2/b)

Tardigrade Admin · Accepted Answer

Consider the free body diagram shown in the figure.
For the equilibrium in the normal direction, we have

T = m g cos θ

In the tangential direction,

m L \ddot{θ} = - m g sin θ - mb v

For small angle

θ

, we have

sin θ \approx θ

and

_{v} = L \dot{θ}

Thus,

m L \ddot{θ} + mb L \dot{θ} + m g θ = 0

Solution to the above differential equation is

θ (t) = θ_{0} e^{- \frac{b t}{2}} sin (ω_{d} t + ϕ)

Let average lifetime be

τ

.
Amplitude at

t = 0

is

θ_{0}

and amplitude at

= τ

is

θ_{0} e^{- \frac{b τ}{2}}

But the ratio is equal to

1/ e

Thus,

e^{- \frac{b τ}{2}} = e^{- 1} \Rightarrow τ = 2/ b

$\frac{0.693}{b}$

$b$

$\frac{1}{b}$

$\frac{2}{b}$

b0.693​

b

b1​

b2​

$\frac{0.693}{b}$

$b$

$\frac{1}{b}$

$\frac{2}{b}$