Lecture on Oscillations and the Millennium Bridge

Introduction

• June 2001: Millennium Bridge in London unveiled.
• Pedestrian bridge spanning the River Thames.
• Swayed dramatically due to footsteps and had to be closed.
• People’s leaning into the swaying exacerbated the issue, causing the bridge to form a giant S shape (horizontal wave).
• Engineers took nearly two years to fix the issue.

Simple Harmonic Motion (SHM)

• Physics concept: oscillations or back-and-forth motion.
• Specific type: simple harmonic motion where oscillations follow a consistent pattern.

Ball and Spring Example for SHM

• Ball attached to a horizontal spring on a table represents SHM.
• At rest: equilibrium state.
• Stretch the spring and release, ball oscillates back and forth.
• Fully theoretical: ignores friction.

Energy in SHM

• Two forms of energy: kinetic energy and potential energy.
• Turning points: points where the spring is fully compressed/stretched (amplitude).
• At turning points: all energy is potential energy.
• As ball moves towards the middle, kinetic energy increases and potential energy decreases.
• At equilibrium point: kinetic energy is max, total energy is conserved.

Maximum Velocity Equation

• Total energy equations combined.
• Maximum velocity (v_max) = amplitude * sqrt(spring constant / mass).*

SHM Properties

• Period, frequency, and angular velocity.
• Relationship between SHM and uniform circular motion (UCM).

Comparison with Uniform Circular Motion (UCM)

• UCM: Marble moving on a ring at constant speed.
• If viewed edge-on, the marble appears to move back and forth.
• Similarities in motion & velocity equations.

Relevant UCM Equations

• Period (T): Time for one complete revolution.
  • T = 2 * pi * amplitude / max velocity.
  • Simplified to T = 2 * pi * sqrt(mass / spring constant).
• Frequency (f): Revolutions per second.
  • f = 1 / T = (1 / 2 * pi) * sqrt(spring constant / mass).
• Angular Velocity (ω): Radians per second.
  • ω = 2 * pi * frequency = sqrt(spring constant / mass).

Position Change Over Time

• Involves trigonometry.
• Cosine of the angle relates to the horizontal distance from the center.
• Position equation: x = amplitude * cos(ωt).*

Practical Application: Millennium Bridge Sway

• Sway caused by oscillation, exacerbated by resonance.
• Resonance: applying force at the right frequency to increase amplitude.
• Engineers accounted only for vertical oscillations, not horizontal swaying.
• Leaning of pedestrians caused horizontal sway, leading to increased resonance.
• Changes applied: counteract oscillations (resonance).

Conclusion

• Lesson on simple harmonic motion: energy, period, frequency, angular velocity.
• Application of theory to real-life example: Millennium Bridge.
• Upcoming topics include wave discussions.