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

2. T x
A=xmax t v t a t

3. 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)!

4. 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 C. 60 D. 100

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

6. 7. Damped Harmonic Motion
x(t) t

7. 8. Resonance

8. 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

9. 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