# PHYSICS OF VIBRATIONS Exercises

1. Write short note on Lissajous figures.

2. What are Lissajous figures? Discuss the formation of these figures, when the periods of         vibration of two simple harmonic motions are in the ratio of 1 : 1

3. What fraction of total energy of a simple harmonic oscillator is kinetic, when its displacement

from the mean position is equal to half the amplitude.

4. What is quality factor?

5. What is the difference between free, damped and forced oscillations? Give the theory of  forced oscillations. Discuss the condition of resonance.

6. What do you mean by resonance? Discuss the sharpness of resonance.

7. What is sample harmonic oscillator? Establish the differential equation for it and solve it to

deduce the expression for velocity, displacement and time period.

8. Show that the solution of equation d2 /dt2 + w2=0  is x = a sin (w + φ), where x0 is the initial

dt displacement, for the initial velocity v0 amplitude a =  √ x20+( v20 /w2)and initial phase angle

φ =tan -1 (x0w/v0)

9. Deduce an expression for potential energy, kinetic energy and total energy for simple          harmonic oscillator and prove that the average kinetic energy is equal to the average potential energy and it is equal to half the total energy.

10. What is a simple pendulum? Show that the oscillations of a simple pendulum for small   amplitude are simple harmonic. Establish the expression for the time period of its oscillations.

11. The displacements of two mutually perpendicular simple harmonic oscillations of same frequency are x = a sin (wt +φ) and y =b sin wt, obtain the expression for the resultant oscillation obtained due to their superposition. What will be the resultant path if φ =00 ,π/2 and π?

12. What is meant by a damped harmonic oscillator? Write the differential equation for it and  find its solution.

13. Explain under damped case and obtain expressions for average total energy and average power loss by it.

14. What do you understand by the relaxation time, damping constant and quality factor of a damped harmonic oscillator? Obtain their expressions.

15. If the damping constant b << n (where n is the angular frequency of the free oscillations). Show that (t) average total energy Eav = 1/2 mw2a20eo-2bt and (ii) power dissipation P = Eav/t

16. Write down the differential equation for a forced harmonic oscillator and explain the significance of each term in it. Obtain solution of this equation and discuss it. Write the condition of resonance and explain the sharpness of resonance.

17. Write down the expression for the amplitude of a forced oscillator. Explain the resonance and

half power points on the amplitude-frequency curve. Show that the quality factor Q = wr/2 ∆w where the symbols have their usual meanings.

18. Write short notes on

(z) Damped harmonic oscillator

(ii) Simple harmonic oscillator

(iii) Forced harmonic oscillator

(iv) Quality factor

(v) Resonance and its sharpness.

19. What is meant by the quality factor of a damped harmonic oscillator? Derive an expression for the same.