A pendulum with time period of 1 s is losing energy due to damping. At a certain time its energy is 45 J. If after completing 15 oscillations, its energy has become 15 J, its damping constant (in s⁻¹) is:

The energy of a pendulum is given as
E₁ = 1/2KA₁² = 45 J
E₂ = 1/2KA₂² = 15 J
Taking the ratio of (i) and (ii):
A₁/A₂ = √3
For a damped oscillator, mẍ + bẋ + cx = 0, the amplitude is given by:
A(t) = A₀e⁻ᵇᵗ/²ₘcos(ωt + φ)
A₁ = A₀e⁻ᵇᵗ/²ₘcos(ωt + φ)
A₂ = A₀e⁻ᵇ(t+15T)/²ₘcos(ωt + φ)
The ratio of amplitudes is:
A₂/A₁ = e⁻ᵇ¹⁵ᵀ/²ₘ
where b/m is the damping constant. t is equal to the time taken for 15 oscillations:
t = 15T = 15 s
where T is the time period for 1 complete oscillation.
A₁/A₂ = eᵇ¹⁵/²ₘ
15b/2m = ln(A₁/A₂)
b/m = (2/15)ln3