1. Intro to Harmonic Oscillator Ankit, Donghun and Masha 4/2/2014 Intended for an intro physics course for non-physics majors (premeds)

2. Learning Objectives You will learn about the natural frequency of a mass-spring harmonic oscillator system. You will be able to predict how changing the mass of a harmonic oscillator affects the frequency. You will be able to use the mass, spring constant, frequency relationship to interpret features of molecular vibration spectrum. You will recognize the relationship between natural frequency and resonance.

3. Q: Consider two oscillators (Q and R) composed of identical springs with masses m and 4m attached to them. How will the natural frequencies of the oscillators compare? A. f Q = f R B. f Q < f R C. f Q > f R D. Can’t tell E. None of the above m m Q 4m R

4. Q: Consider two oscillators (Q and R) composed of identical springs with masses m and 4m attached to them. How will the natural frequencies of the oscillators compare? A. f Q = f R B. f Q = 4f R C. f Q = ¼f R D. f Q = 2f R E. f Q = ½f R m m Q 4m R

5. How will the natural frequencies of the oscillators compare? Answer: D. f Q = 2f R m m Q 4m R (Hz) (N/m) (kg)

6. Discussion question What do you think will happen if you jiggle the mass at a frequency close to the natural frequency? – Give it a try! Discuss what you think is happening with your group.

7. Resonance examples Music All musical instruments produce sound via a vibrating element In a violin, the differing thicknesses of the strings result in different resonant frequencies The musician touches the instrument to change the resonance, ie pitch, of the sound played Architecture Bridges, tall buildings and other structures are susceptible to collapse when driven at their natural frequency by external factors like wind or people walking Everyday phenomena Swings Wine glass vibration http://education-portal.com/academy/lesson/resonance-definition-transmission-of-waves.html#lesson http://www.kshitij-school.com/Study-Material/Class-11/Physics/Superposition-and-standing-waves/Resonance-standing- waves.aspx Tacoma Narrows Bridge

8. Diatomic molecule Note: take much heavier Cl atom to be stationary (like a wall) Cl m (H or D)

9. Vibrational energy levels HClDCl n=0 n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=0 n=1 n=2 n=3 n=4 n=5 n=6 n=7 Energy

10. D. E. Mann et al., J. Chem. Phys. 44 3453 (1966) HCl f = 8.555 x 10 13 Hz Absorption (a.u.) Frequency (Hz) DCl f = 6.216 x 10 13 Hz Vibrational Spectrum of HCl and DCl

11. From http://hyperphysics.phy-astr.gsu.edu HCl D. E. Mann et al., J. Chem. Phys. 44 3453 (1966) H 35 Cl f = 8.555 x 10 13 Hz H 37 Cl Absorption (a.u.) Frequency (Hz) D 35 Cl f = 6.216 x 10 13 Hz D 37 Cl Vibrational Spectrum of HCl and DCl

12. Vacuum Water f ? Air Next Class: Damped Harmonic Oscillator

13. Homework: diatomic molecule with similar masses m1m1 m2m2 m1m1 m2m2