# if a simple pendulam has significant amplitude ( upto a factor of 1/e of original) only in the period between t=0 to t=τ second  ,then may be called the average life of pendulum.when the spherical bob of the pendulum retardation (due to viscous drag) proportional to its velocity with b as the constant of proportionality , the average time of the pendulum is (assuming damping is small) in seconds-

christy
29 Points
4 years ago
For damped harmonic motion,
ma = -kx – mbv
or
ma + mbv + kx = 0
Solution to above equation is
x = A0e-bt/2m sin ωt ; with ω2 = k/m – b2/4
where amplitude drops exponentially with time
AT = A0e-bT/2
Average time T is the duration when amplitude drops by 63%. i.e., it becomes A0/e
Thus,
AT = A0/e = A0e-bt/2
or bt/2 = 1 or T = 2/b