Intro to Harmonic Oscillator

Ankit, Donghun and Masha4/2/2014

Intended for an intro physics course for non-physics majors (premeds)

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.

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. fQ = fR

B. fQ < fR

C. fQ > fR

D. Can’t tell E. None of the above

mQ

4mR

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. fQ = fR

B. fQ = 4fR

C. fQ = ¼fR

D. fQ = 2fR

E. fQ = ½fR

mQ

4mR

How will the natural frequencies of the oscillators compare?

Answer: D. fQ = 2fR

mQ

4mR𝒇=𝟏𝟐𝝅 √ 𝒌

𝒎(Hz)

(N/m)

(kg)

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.

Resonance examplesMusic• 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#lessonhttp://www.kshitij-school.com/Study-Material/Class-11/Physics/Superposition-and-standing-waves/Resonance-standing-waves.aspx

Tacoma Narrows Bridge

Diatomic molecule

Note: take much heavier Cl atom to be stationary (like a wall)

Cl m (H or D)

Vibrational energy levels

HCl DCl

n=0

n=1

n=2

n=3

n=4

n=5

n=6

n=7

n=0

n=1n=2n=3n=4n=5n=6n=7

Ener

gy

D. E. Mann et al., J. Chem. Phys. 44 3453 (1966)

HCl f = 8.555 x 1013 Hz

Abso

rptio

n (a

.u.)

Frequency (Hz)

DCl f = 6.216 x 1013 Hz

Vibrational Spectrum of HCl and DCl

cf. f = 6.049 x 1013 Hz

From http://hyperphysics.phy-astr.gsu.edu

HCl

D. E. Mann et al., J. Chem. Phys. 44 3453 (1966)

H35Cl f = 8.555 x 1013 Hz

H37Cl

Abso

rptio

n (a

.u.)

Frequency (Hz)

D35Cl f = 6.216 x 1013 Hz

D37Cl

Vibrational Spectrum of HCl and DCl

Vacuum Water

f ? f ? f ?

Air

Next Class: Damped Harmonic Oscillator

Homework: diatomic molecule with similar masses

m1 m2 m1 m2

m1 m2 m1 m2