JEE Physics Oscillations Notes Set 06

Free, Damped and Driven Oscillations

An ideal oscillator keeps its energy forever. A real one loses energy to friction or air drag, so its amplitude shrinks; if an outside agent keeps pushing it periodically, the oscillation is sustained. Three situations matter: free oscillation (no drag, no push), damped oscillation (drag only) and forced or driven oscillation (drag plus a periodic push).

Damped Oscillations

For small speeds the drag is proportional to the velocity and opposite to it, \( F_d=-bv \), where b depends on the medium and the shape of the body. For a block on a spring the equation of motion is

\[ m\frac{d^2x}{dt^2}+b\frac{dx}{dt}+kx=0 \]

Trying \( x=e^{rt} \) gives \( mr^2+br+k=0 \) with roots \( r=\dfrac{-b\pm\sqrt{b^2-4mk}}{2m} \). The sign of \( b^2-4mk \) decides the type of motion.

Underdamped Motion

1. Displacement: \( x=Ae^{-bt/2m}\cos(\omega't+\phi) \), where \( \omega'=\sqrt{\dfrac km-\dfrac{b^2}{4m^2}} \).
2. The amplitude falls exponentially, as the envelope \( Ae^{-bt/2m} \). Equal time intervals reduce the amplitude by the same factor.
3. For light damping \( \omega'\approx\omega=\sqrt{k/m} \), so the period is almost unchanged.
4. Energy \( E=\tfrac12kA^2e^{-bt/m} \): it decays twice as fast as the amplitude, because it depends on the square of the amplitude.
5. Time for the amplitude to halve: \( t_{1/2}=\dfrac{2m\ln2}{b} \); for the energy to halve it is \( \dfrac{m\ln2}{b} \).
6. The quality factor is \( Q=2\pi\dfrac{E}{|\Delta E|}\approx\dfrac{\omega}{b/m} \) (energy lost per cycle \( |\Delta E|=2\pi E/Q \)); an undamped oscillator has \( Q\to\infty \).

Forced Oscillations and Resonance

Let a periodic force \( F=F_0\cos\omega_dt \) act on a damped oscillator of natural angular frequency \( \omega \). The equation is \( m\ddot x+b\dot x+kx=F_0\cos\omega_dt \). After the free oscillation dies out, the body oscillates at the driving frequency, not its natural one:

\[ x=A\cos(\omega_dt+\phi),\qquad A=\frac{F_0}{\sqrt{m^2(\omega^2-\omega_d^2)^2+\omega_d^2b^2}} \]

1. Driving frequency far from natural frequency with small damping: \( A\approx\dfrac{F_0}{m|\omega^2-\omega_d^2|} \).
2. Driving frequency close to the natural frequency: the first term in the root is negligible and \( A\approx\dfrac{F_0}{\omega_db} \). The amplitude is large but finite; it would be infinite only with no damping.
3. This large increase of amplitude when the driving frequency matches the natural frequency is resonance. A child on a swing is a familiar case: pushing in step with the swing builds up the amplitude.
4. Smaller damping gives a taller and narrower resonance peak; heavier damping flattens it and lowers the peak.

Worked Examples

Example 1. A damped oscillator has m = 200 g, k = 90 N/m and b = 40 g/s. Find the period, the time for the amplitude to halve and the time for the energy to halve.

1. \( \omega'=\sqrt{90/0.2-(0.04)^2/(4\times0.04)}=\sqrt{450-0.01}=21.2 \) rad/s, so \( T=0.30 \) s.
2. \( t_{1/2}=\dfrac{2(0.2)\ln2}{0.04}=6.93 \) s for the amplitude.
3. For the energy it is \( \dfrac{0.2\ln2}{0.04}=3.47 \) s, half of the above.

Example 2. Show that \( \ddot x+\dot x+3x=0 \) is underdamped and find the damped angular frequency.

1. m = 1, b = 1, k = 3, so \( b^2=1<4mk=12 \): underdamped.
2. \( \omega_d=\sqrt{3-\tfrac14}=\dfrac{\sqrt{11}}2 \) rad/s.

Example 3. Classify (i) \( \ddot x+4\dot x+3x=0 \) and (ii) \( \ddot x+4\dot x+4x=0 \).

1. (i) \( b^2=16>12 \): overdamped.
2. (ii) \( b^2=16=4mk \): critically damped.

Example 4. The amplitude of a damped oscillator falls to 0.9 of its value in 5 s. What fraction remains after 15 s?

1. \( e^{-5\gamma}=0.9 \), so after 15 s the factor is \( 0.9^3=0.729 \).

Quick Revision Points

1. Damping force \( -bv \); amplitude decays as \( e^{-bt/2m} \), energy as \( e^{-bt/m} \).
2. Underdamped if \( b^2\lt 4mk \), critical if equal, overdamped if greater.
3. A driven oscillator vibrates at the driving frequency; its amplitude peaks near the natural frequency (resonance).
4. Damping limits the resonance amplitude to about \( F_0/\omega_db \).