4. Harmonic oscillations

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Presentation transcript:

4. Harmonic oscillations Relation between circular motion and simple harmonic oscillations y R=A x Period of SHO is independent from amplitude!

T x A=xmax t v t a t

5. Hook’s law and simple harmonic motion Newton’s second law: Simple harmonic motion: Question: Two identical masses hang from two identical springs. In case 1, the mass is pulled down 2 cm and released. In case 2, the mass is pulled down 4 cm and released. How do the periods of their motions compare? A. T1 < T2 B. T1 = T2 C. T1 > T2 Period is independent of amplitude! This is in fact general to all SHM (not only for springs)!

Question: Mass m attached to a spring with a spring constant k. If the mass m increases by a factor of 4, the frequency of oscillation of the mass __. is doubled is multiplied by a factor of 4 is halved is multiplied by a factor of 1/4 Question: A 2.0 kg mass attached to a spring with a spring constant of 200 N/m. The angular frequency of oscillation of the mass is __ rad/s. A. 2 B. 10 C. 60 D. 100

6. Energy in the simple harmonic motion U E x –A A K t U t E t Total mechanical energy is constant through oscillation: conservation of energy!

7. Damped Harmonic Motion x(t) t

8. Resonance

9. Two-mass vibrator Longitudinal vibration Modes (a) and (b) are independent Transverse vibration m m Violin string normally vibrates transversely Two-mass vibrator has 2 longitudinal and 4 transverse modes of vibration

10. Systems with many modes N-mass vibrator has N longitudinal and 2N transverse modes of vibration Vibrating string can be thought of as the limit of the mass-spring system when the number of masses becomes very large 11. Vibrations in musical instruments Strings Membrane Bar Plate Tuning fork 12. Vibration Spectra