Question

If a simple pendulum has significant amplitude up to a factor of original only in the period between then it may be called the average life of the pendulum

Answer 1

As retardation=bv
retarding force=mbv
net restoring torque when angular displacement is θ is,
τ=−mgl sinθ+mbvl
Iα=−mgl sinθ+mbvl   ........[because τ=Iα]  ........ (1)
Also, I=ml2  ... (2)
from (1) & (2),
d2θ/dt2=α=−glsinθ+bvlfor small damping, the solution of the above differential equation will be