Introduction
This is a damped harmonic oscillator problem where we need to calculate multiple physical parameters based on the values of mass, damping constant, and spring constant. The motion is governed by the differential equation:
m d²x/dt² + γ dx/dt + kx = 0
Given:
- m = 0.50 kg
- γ = 0.70 kg/s
- k = 70 N/m
Step 1: Calculate Angular Frequency and Period
Natural angular frequency (without damping):
ω₀ = √(k/m) = √(70 / 0.5) = √140 ≈ 11.83 rad/s
Damping coefficient:
β = γ / (2m) = 0.70 / (2 × 0.5) = 0.7
Damped angular frequency:
ω_d = √(ω₀² – β²) = √(140 – 0.49) ≈ √139.51 ≈ 11.81 rad/s
Period of damped motion:
T = 2π / ω_d ≈ 2π / 11.81 ≈ 0.532 s
Step 2: Number of Oscillations for Amplitude to Halve
The amplitude decays exponentially as A(t) = A₀ e-βt
We want A(t) = A₀ / 2 → e-βt = 1/2
t = ln(2) / β ≈ 0.693 / 0.7 ≈ 0.99 s
Number of oscillations = t / T = 0.99 / 0.532 ≈ 1.86 oscillations
Step 3: Number of Oscillations for Energy to Halve
Mechanical energy E(t) decays as: E(t) = E₀ e-2βt
Set: E(t) = E₀ / 2 → e-2βt = 1/2
t = ln(2) / (2β) = 0.693 / (2 × 0.7) ≈ 0.495 s
Number of oscillations = t / T = 0.495 / 0.532 ≈ 0.93 oscillations
Step 4: Relaxation Time (τ)
τ = 1 / β = 1 / 0.7 ≈ 1.43 s
Step 5: Quality Factor (Q)
Q = ω₀ / (2β) = 11.83 / (2 × 0.7) ≈ 8.45
Final Answers
- (i) Period of motion: 0.532 s
- (ii) Number of oscillations for amplitude to halve: 1.86
- (iii) Number of oscillations for energy to halve: 0.93
- (iv) Relaxation time: 1.43 s
- (v) Quality factor: 8.45
Conclusion
This problem illustrates the effect of damping on harmonic motion. By using the standard equations for damped oscillations, we evaluated how quickly the amplitude and energy decay, and how efficiently the system sustains oscillations through the quality factor.