Summary

There are two physical effects at play in a damped oscillator. The first is the damping which tries to bring any motion to a stop. This operates on a time-scale $T_d \approx 1/\beta$. The restoring force exerted by the spring tries to make the system oscillate and this operates on a time-scale $T_0=1/\omega_0$. We have overdamped oscillations if the damping operates on a shorter time-scale compared to the oscillations ie. $T_d < T_0$ which completely destroys the oscillatory behaviour.

Figure 2.4 shows the behaviour of a damped oscillator under different combinations of damping and restoring force. The plot is for $\omega_0=1$, it can be used for any other value of the natural frequency by suitably scaling the values of $\beta$. It shows how the decay rate for the two exponentially decaying overdamped solutions varies with $\beta$. Note that for one of the modes the decay rate tends to zero as $\beta$ is increased. This indicates that for very large damping a particle may get stuck at a position away from equilibrium.

Figure 2.4:

Problems

1. Obtain solution (2.20) for critical damping as a limiting case $(\beta\rightarrow \omega_0)$ of overdamped solution (2.18).

2. Find out the conditions for the initial displacement $x(0)$ and the initial velocity $\dot{x}(0)$ at $t=0$ such that an overdamped oscillator crosses the mean position once in a finite time.

3. An under-damped oscillator has a time period of $2 {\rm s}$ and the amplitude of oscillation goes down by $10 \%$ in one oscillation. [a.] What is the logarithmic decrement $\lambda$ of the oscillator? [b.] Determine the damping coefficient $\beta$. [c.] What would be the time period of this oscillator if there was no damping? [d.] What should be $\beta$ if the time period is to be increased to $4 {\rm s}$? ([a.] $1.05 \times 10^{-1}$ [b.] $2.7 \times 10^{-2} {\rm s}^{-1}$ [c.]$2 {\rm s}$ [d.] $2.72 {\rm s}^{-1}$
4. Two identical under-damped oscillators have damping coefficient and angular frequency $\beta$ and $\omega$ respectively. At $t=0$ one oscillator is at rest with displacement $a_0$ while the other has velocity $v_0$ and is at the equilibrium position. What is the phase difference between these two oscillators. ( $\pi/2 -\tan^{-1} (\beta/\omega)$)
5. A door-shutter has a spring which, in the absence of damping, shuts the door in $0.5 {\rm s}$. The problem is that the door bangs with a speed $1 {\rm m}/{\rm s}$ at the instant that it shuts. A damper with damping coefficient $\beta$ is introduced to ensure that the door shuts gradually. What are the time required for the door to shut and the velocity of the door at the instant it shuts if $\beta=0.5 \pi$ and ? Note that the spring is unstretched when the door is shut. ($0.57 {\rm s}$, $4.67 \times10^{-1} {\rm m}/ {\rm s}$; , $8.96 \times 10^{-2} {\rm m}/{\rm s}$)
6. An $LCR$ circuit has an inductance $L=1 \, {\rm mH}$, a capacitance $C=0.1 \, \mu {\rm F}$ and resistance $R=250 \Omega$ in series. The capacitor has a voltage $10 \, {\rm V}$ at the instant $t=0$ when the circuit is completed. What is the voltage across the capacitor after $10 \mu {\rm s}$ and $20 \mu {\rm s}$? ( $7.64 \, {\rm V}$, $4.84 \, {\rm V}$ )
7. A highly damped oscillator with $\omega_0= 2 \, {\rm s}^{-1}$ and $\beta=10^4 \, {\rm s}^{-1}$ is given an initial displacement of $2 \, {\rm m}$ and left at rest. What is the oscillator's position at $t=2 \, {\rm s}$ and $t=10^4 \, {\rm s}$? ( $2.00 \, {\rm m}$, $2.70 \times 10^{-1} \, {\rm m}$)
8. A critically damped oscillator with $\beta = 2 \, {\rm s}^{-1}$ is initially at $x=0$ with velocity $6 \, {\rm m}\, {\rm s}^{-1}$. What is the furthest distance the oscillator moves from the origin? ( $1.10 \, {\rm m}$)
9. A critically damped oscillator is initially at $x=0$ with velocity $v_0$. What is the ratio of the maximum kinetic energy to the maximum potential energy of this oscillator? ($e^2$)
10. An overdamped oscillator is initially at $x=x_0$. What initial velocity, $v_0$, should be the given to the oscillator that it reaches the mean position (x=0) in the minimum possible time.
11. We have shown that the general solution, $x(t)$, with two constants can describe the motion of damped oscillator satisfying given initial conditions. Show that there does not exist any other solution satisfying the same initial conditions.