If a simple pendulum has the significant amplitude (up to a factor of 1/e of original) only in the period between t = Os to t = τs, 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

Solution

2/b

For damped harmonic motion,
ma = - kx - mbv
or
ma + mbv +kx = 0
Solution to above equation is
$straight x space equals space straight A subscript straight o straight e to the power of negative bt over 2 end exponent space sin space ωt semicolon space space with space straight omega squared space equals space straight k over straight m minus straight b squared over 4$
Where amplitude drops exponentially with time
$straight A subscript straight tau space equals space straight A subscript straight o straight e to the power of negative bτ over 2 end exponent$
Average time τ is that duration when amplitude drops by 63%. i.e., becomes A_o/e
Thus,
$straight A subscript straight tau space equals space straight A subscript straight o over straight e space equals space straight A subscript straight o straight e to the power of negative bt over 2 end exponent or space bτ over 2 space equals 1 space or space straight tau space equals 2 over straight b$

Some More Questions From Mechanical Properties of Fluids Chapter

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Mechanical Properties Of Fluids

Some More Questions From Mechanical Properties of Fluids Chapter

What determines the natural frequency of an oscillator?

Is there any relation between uniform circular motion and simple harmonic motion?

What is periodic motion?

Write down the expression for the instantaneous velocity and acceleration of particle vibrating in simple harmonic motion.

How do you define the mean position of particle executing simple harmonic motion?

Define amplitude.

What is the net displacement of the particle executing simple harmonic motion in one complete oscillation?

Are all the periodic motions simple harmonic?

Does the time period of particle vibrating in simple harmonic motion depend upon its displacement?

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