Damped and Forced Oscillations - UNSOLVED PRACTICE SET

Q7. What is damping? Name three examples of damped oscillations from everyday life.

Q8. Distinguish between free oscillations, damped oscillations, and forced oscillations with one example of each.

Q9. What is meant by the relaxation time of a damped oscillator? How is it related to the damping constant?

Q10. Explain what happens to the amplitude of a forced oscillator when:

(i) The driving frequency is much less than the natural frequency

(ii) The driving frequency is much greater than the natural frequency

(iii) The driving frequency equals the natural frequency

Q11. Why are shock absorbers in vehicles designed to provide critical damping rather than underdamping or overdamping?

Q12. A tuning fork stops vibrating after some time when left in air. What type of oscillation is this? What causes the tuning fork to stop?

Q13. Write the differential equation for a damped harmonic oscillator: d²x/dt² + 2b(dx/dt) + ω₀²x = 0. Discuss the three cases of damping:

(i) Underdamped (b < ω₀)

(ii) Critically damped (b = ω₀)

(iii) Overdamped (b > ω₀)

For each case, describe the nature of the solution and sketch the displacement-time graph. Give one practical example for each case.

Q14. Discuss forced oscillations and resonance. Write the differential equation for a forced oscillator and explain the terms. Derive the expression for the amplitude of steady-state forced oscillations. Show that the amplitude is maximum when the driving frequency equals the natural frequency (for small damping). Explain the phenomenon of resonance with examples.

Q15. A student sets up a simple pendulum in a container of water and observes that the oscillations die out quickly.

(i) What type of damping is this? Explain.

(ii) How would the time period compare to the same pendulum in air?

(iii) If the student replaces water with oil, how would the damping change?

(iv) Design an experiment to measure the damping constant for this system.

Q16. A damped oscillator has mass m = 0.5 kg, spring constant k = 50 N/m, and damping constant b = 0.5 s⁻¹.

(i) Calculate the natural angular frequency ω₀.

(ii) Calculate the angular frequency of the damped oscillation ω.

(iii) Calculate the time in which the amplitude reduces to 1/e of its initial value.

(iv) If the initial amplitude is 10 cm, calculate the amplitude after 10 oscillations.

(v) How many oscillations occur before the amplitude drops to 1% of its initial value?

Q17. A car of mass 1000 kg is supported by four springs, each with force constant 20,000 N/m. The shock absorbers provide a damping constant such that the system is critically damped.

(i) Calculate the natural frequency of oscillation of the car.

(ii) Calculate the damping constant required for critical damping.

(iii) If the car hits a bump and is displaced 5 cm, how long does it take to return to within 1 mm of equilibrium?

(iv) Why is critical damping preferred in vehicle suspension systems?

Q18. In a school physics demonstration, a pendulum of length 1 m is driven by a motor that applies a periodic force. The natural frequency of the pendulum is f₀.

(i) Calculate the natural frequency of the pendulum.

(ii) The motor drives the pendulum at frequencies 0.5f₀, f₀, and 2f₀. Describe what happens to the amplitude in each case.

(iii) If the damping in the system is increased, how does the resonance curve change? Sketch the amplitude vs driving frequency for light damping and heavy damping.

(iv) Explain why soldiers are asked to break step while marching across a bridge.

(Given: g = 9.8 m/s²)