# Lecture 1: Periodic Phenomena

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## Lecture Topics

 Periodic Phenomena (Oscillations, Waves) Simple Harmonic Oscillators Complex Notation Differential Equations Physical Pendulum

## Learning Objectives

By the end of this lecture, you should:

• recognize examples of periodic motion in everyday life.
• see some complicated oscillatory motion and production of sound through oscillatory motion.
• know that sound is pressure oscillations in air of frequency 20 Hz to 20 kHz.
• explain simple harmonic oscillation (SHO) through its basic equation, relation to circular motion, and complex exponential form.
• relate phase, angular velocity, frequency, and period.
• explain the dynamics of springs qualitatively and quantitatively.
• understand the relation of complex numbers to circular motion.
• quantitatively explain the dynamics of the pendulum in the small (angle) amplitude approximation.
• understand the concept of wave polarization.

## Check Yourself

• If the speed of sound in air is 1236 km/hour, what are the respective wavelengths of the highest (20 kHz) and lowest (20 Hz) frequency sound waves a human can hear?

0.0172 m; 17.2 m

• For BOTH the cases of a pendulum and a mass on a spring, choose respectively the unit calculations that show that the period formula gives a result with the units of time.

$\frac{l}{g} \sf{\mbox{has units}} \frac{[m]}{[m/{{s}^{2}}]}=\frac{[1]}{[1/{{s}^{2}}]}=[{{s}^{2}}]$

$\frac{m}{k} \sf{\mbox{has units}} \frac{[kg]}{[N/m]}=\frac{[kg]}{[kg\cdot m/{{s}^{2}}/m]}=[{{s}^{2}}]$

The only further operation is a square root which gives [s].

• Again consider the pendulum and the mass on a spring, and describe how the angular frequency formula matches your experience and intuition.

$\omega =\sqrt{\frac{g}{l}}$

makes sense since we would think the higher restoring force from a higher “g” would make a pendulum go faster, while a longer pendulum is know to vibrate more slowly

$\omega =\sqrt{\frac{k}{m}}$

makes sense since a stiffer spring vibrates with higher frequency, while a larger mass being pushed by a spring moves ponderously.

• For North American AC electricity with a frequency of 60 Hz, what is the angular frequency, including units?