When the Millennium Bridge in London opened in 2000, pedestrians noticed a severe swaying motion. As people walked across, their natural footfalls created small lateral forces. By chance, the frequency of these collective footfalls matched the natural lateral resonant frequency of the bridge structure. This caused the bridge's swaying amplitude to grow dramatically—a classic case of driven, undamped resonance. Engineers had to retrofit the bridge with massive tuned mass dampers (shock absorbers) to dissipate the vibrational energy and prevent destructive oscillations.

Earthquakes release immense energy that travels through the Earth's crust as waves. Seismologists track two primary types: P-waves (Primary) and S-waves (Secondary). P-waves are longitudinal (compressional) waves, traveling faster and pushing/pulling rock in the direction of travel. S-waves are transverse (shear) waves, traveling slower and shaking rock side-to-side perpendicular to the travel direction. The time difference between the arrival of the fast P-waves and the slower S-waves at a seismograph station allows scientists to pinpoint exactly how far away the earthquake epicenter is located.

A $2.0 \text{ kg}$ mass is attached to a horizontal spring with a spring constant $k = 50 \text{ N/m}$. The mass is displaced $0.1 \text{ m}$ from equilibrium and released from rest. What is the period and frequency of the oscillation?

An astronaut on the Moon wants to create a simple pendulum that has a period of exactly $2.0 \text{ seconds}$. If the acceleration due to gravity on the Moon is $1.62 \text{ m/s}^2$, how long must the pendulum string be?

A $0.5 \text{ kg}$ block attached to a spring ($k = 200 \text{ N/m}$) oscillates on a frictionless horizontal surface. If its maximum speed (velocity at equilibrium) is $3.0 \text{ m/s}$, what is the amplitude of the oscillation?

A radio station broadcasts an FM radio wave (an electromagnetic wave traveling at the speed of light, $c = 3.0 \times 10^8 \text{ m/s}$) at a frequency of $95.5 \text{ MHz}$. What is the wavelength of the radio wave?

A $2.0 \text{ m}$ long guitar string has a mass of $0.005 \text{ kg}$. It is pulled tight with a tension of $400 \text{ N}$. If the string is plucked, how fast does the transverse wave pulse travel along the string?

An organ pipe that is open at both ends has a fundamental frequency (first harmonic) of $250 \text{ Hz}$. If the speed of sound in air is $340 \text{ m/s}$, how long is the pipe? What is the frequency of its second harmonic?