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Oscillations and Waves - Free, Damped and Forced Oscillations

Grade 11ICSEPhysics

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Free Oscillations: These occur when a system oscillates with its natural frequency f0f_0 in the absence of any external resistive force or periodic force. The amplitude remains constant over time.

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Natural Frequency: The frequency with which a body oscillates when it is displaced from its mean position and left to itself. For a simple pendulum, f=12πglf = \frac{1}{2\pi}\sqrt{\frac{g}{l}}.

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Damped Oscillations: Oscillations in which the amplitude decreases gradually with time due to dissipative forces like friction or air resistance. The energy of the system is lost as heat to the surroundings.

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Damping Force: It is proportional to the velocity of the particle, expressed as Fd=−bvF_d = -b v, where bb is the damping constant.

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Forced Oscillations: When a body oscillates under the influence of an external periodic force, its oscillations are called forced oscillations. The body eventually oscillates with the frequency of the external force rather than its natural frequency.

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Resonance: A special case of forced oscillations where the frequency of the external periodic force is equal to the natural frequency of the system. This results in a rapid increase in the amplitude of oscillation.

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Sharpness of Resonance: It describes how quickly the amplitude falls off as the external frequency moves away from the natural frequency. It depends on the damping constant bb; smaller damping leads to sharper resonance.

📐Formulae

Natural Angular Frequency: ω0=km\text{Natural Angular Frequency: } \omega_0 = \sqrt{\frac{k}{m}}

Equation of Damped Motion: md2xdt2+bdxdt+kx=0\text{Equation of Damped Motion: } m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0

Displacement in Damped Oscillation: x(t)=Ae−bt2mcos⁡(ω′t+ϕ)\text{Displacement in Damped Oscillation: } x(t) = A e^{-\frac{bt}{2m}} \cos(\omega' t + \phi)

Damped Angular Frequency: ω′=km−b24m2\text{Damped Angular Frequency: } \omega' = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}

Amplitude of Forced Oscillation: A=F0m2(ω02−ωd2)2+b2ωd2\text{Amplitude of Forced Oscillation: } A = \frac{F_0}{\sqrt{m^2(\omega_0^2 - \omega_d^2)^2 + b^2 \omega_d^2}}

Resonance Condition: ωd=ω0\text{Resonance Condition: } \omega_d = \omega_0

💡Examples

Problem 1:

A spring-mass system has a mass m=0.5 kgm = 0.5\text{ kg} and a spring constant k=200 N/mk = 200\text{ N/m}. If the damping constant bb is 0.2 kg/s0.2\text{ kg/s}, calculate the natural frequency and the damped frequency of the system.

Solution:

  1. Natural angular frequency: ω0=km=2000.5=400=20 rad/s\omega_0 = \sqrt{\frac{k}{m}} = \sqrt{\frac{200}{0.5}} = \sqrt{400} = 20\text{ rad/s}.
  2. Natural frequency: f0=ω02π=202π≈3.18 Hzf_0 = \frac{\omega_0}{2\pi} = \frac{20}{2\pi} \approx 3.18\text{ Hz}.
  3. Damped angular frequency: ω′=ω02−(b2m)2=202−(0.22×0.5)2=400−0.04=399.96≈19.999 rad/s\omega' = \sqrt{\omega_0^2 - (\frac{b}{2m})^2} = \sqrt{20^2 - (\frac{0.2}{2 \times 0.5})^2} = \sqrt{400 - 0.04} = \sqrt{399.96} \approx 19.999\text{ rad/s}.

Explanation:

The natural frequency is determined solely by the mass and spring constant. The damping constant reduces the frequency of oscillation, though in many mechanical systems with low damping, ω′≈ω0\omega' \approx \omega_0.

Problem 2:

A forced oscillator is driven by an external force F=F0cos⁡(ωdt)F = F_0 \cos(\omega_d t). At what frequency ωd\omega_d does resonance occur, and what happens to the amplitude if the damping constant bb is very small?

Solution:

Resonance occurs when the driving frequency matches the natural frequency, i.e., ωd=ω0\omega_d = \omega_0. The amplitude is given by A=F0m2(ω02−ωd2)2+b2ωd2A = \frac{F_0}{\sqrt{m^2(\omega_0^2 - \omega_d^2)^2 + b^2 \omega_d^2}}. At ωd=ω0\omega_d = \omega_0, the formula simplifies to Amax=F0bω0A_{max} = \frac{F_0}{b \omega_0}.

Explanation:

If the damping constant bb is very small, the denominator becomes very small at resonance, leading to a very large (theoretically infinite if b=0b=0) amplitude. This is why resonance can be destructive in structures like bridges.

Free, Damped and Forced Oscillations Revision - Class 11 Physics ICSE