13.
A particle of mass m undergoes harmonic oscillations with period T, about x = 0. Now a
force F is applied to the particle, acting opposite in direction to its instantaneous velocity:
F = – kv. What happens when the particle is released from rest from some x > 0:
a. It undergoes a steady oscillation with a period larger than T
b. It undergoes a decaying oscillation
c. It no longer oscillates, it moves in one direction with decreasing speed
d. It could oscillate or not, depending on parameter values
Damped Harmonic Oscillator: Particle Released from Rest with Viscous Drag F = -kv
A particle in simple harmonic motion (SHM) with period T about x=0 now experiences an additional viscous damping force F = -kv opposing its velocity. Released from rest at x > 0, the system becomes a damped harmonic oscillator. The correct answer is b. It undergoes a decaying oscillation.
Equation of Motion
The original SHM equation is m d²x/dt² + k x = 0, where T = 2π √(m/k) and ω₀ = √(k/m). Adding damping gives:
Here k is the damping coefficient. Define γ = k/(2m) and discriminant D = k² – 4 m k_spring.
Solution Behavior
Characteristic roots are r = [-k ± √D] / (2m). For typical light damping where D < 0 (underdamped case, k < 2 √(m k_spring)), the solution is:
with ω_d = √(ω₀² – γ²). Released from rest at x(0) = x₀ > 0, v(0) = 0 yields oscillatory motion with exponentially decaying amplitude.
Option Analysis
- a. Steady oscillation with period larger than T: Incorrect. Damping dissipates energy; no steady state without driving force. Period increases slightly to 2π/ω_d > T, but amplitude decays.
- b. Decaying oscillation: Correct for underdamped case (γ < ω₀). Particle oscillates about x=0 with decreasing amplitude until rest.
- c. No oscillation, moves one direction with decreasing speed: Wrong. Spring force -kx pulls back toward x=0 from x>0; damping slows but restoring force causes crossing.
- d. Could oscillate or not depending on parameters: Partially true (overdamped: no oscillation; underdamped: yes), but question implies standard physics context (underdamped) without specified heavy damping.
Damping Regimes Table
| Regime | Condition (D) | Motion from x>0, v=0 | Period |
|---|---|---|---|
| Underdamped | D < 0 | Oscillates, decays | > T |
| Critically damped | D = 0 | Returns fastest, no oscillation | None |
| Overdamped | D > 0 | Returns slowly, no oscillation | None |
In exams like CSIR NET, “harmonic oscillations with added drag” assumes underdamped decaying oscillation unless specified otherwise.
