Study Notes on Oscillations and Simple Harmonic Motion

Introduction to the Millennium Bridge Incident

In June 2001, a significant engineering endeavor was unveiled in London - the Millennium Bridge, a pedestrian bridge over the River Thames. While the bridge was aesthetically appealing and expected to serve a practical purpose, it was closed shortly after its opening due to unforeseen issues arising from people's usage.

Bridge Failure Due to Swaying

Upon pedestrian use, the Millennium Bridge began to sway dramatically back and forth, primarily influenced by the force generated by footsteps. As individuals walked, they instinctively leaned into the bridge's oscillation, inadvertently exacerbating the swaying. Eventually, this motion intensified to the point where the bridge contorted into a shape resembling a giant 'S' or horizontal wave. This severe oscillation prompted the immediate closure of the bridge, requiring engineers nearly two years to rectify the situation.

Understanding Oscillations

The fundamental issue with the Millennium Bridge’s swaying lies in the concept of oscillations, specifically simple harmonic motion (SHM), which is characterized by back-and-forth motion in a predictable pattern.

  • Simple Harmonic Motion: This type of motion can be exemplified by observing a ball attached to a horizontal spring, positioned at rest on a surface (equilibrium). When the ball is displaced (stretched or compressed), it will engage in oscillating motion when released, ideally in a frictionless environment.

Energy Dynamics in Simple Harmonic Motion

To fully grasp the behavior of the oscillating ball, one must comprehend its energy states:

  1. Kinetic Energy (KE): This is the energy attributed to the motion of the ball. During its oscillation, there are two key points referred to as the turning points where the ball has zero kinetic energy due to its temporary state of rest (fully compressed or stretched spring).
  2. Potential Energy (PE): At these turning points, the ball's energy is entirely potential energy, calculable as: PE = \frac{1}{2} k A^2 where:
    • k = spring constant
    • A = amplitude (maximum displacement from equilibrium)
  3. Total Energy: Throughout the motion, the total energy remains conserved, oscillating between kinetic and potential forms. Specifically, as the ball approaches the center (equilibrium), kinetic energy increases while potential energy decreases.
  4. Maximum Kinetic Energy: At the equilibrium position, potential energy drops to 0, and the kinetic energy reaches its peak, defined by the equation: KE{max} = \frac{1}{2} m v{max}^2 where:
    • m = mass of the object
    • v_max = maximum velocity

Calculating Maximum Velocity

To determine the maximum velocity of the ball, we can derive a formula by equating the maximum potential energy to maximum kinetic energy, leading to:
v_{max} = A \sqrt{\frac{k}{m}}
This equation provides crucial insight into the maximum velocity achievable by the oscillating ball on the spring.

Properties of Simple Harmonic Motion

In addition to energy and velocity, the ball exhibits key properties such as:

  • Period (T): The time taken for one complete cycle of motion. It can be calculated as:
    T = 2\pi \sqrt{\frac{m}{k}}
  • Frequency (f): The number of oscillations per unit time, inversely related to the period:
    f = \frac{1}{T} = \frac{1}{2 \pi} \sqrt{\frac{k}{m}}
  • Angular Velocity (\omega): Measured in radians per second, expressed as:
    \omega = 2\pi f = \sqrt{\frac{k}{m}}
    These properties make clear connections with related concepts in physics, particularly the analogy to uniform circular motion.

Comparison to Uniform Circular Motion

Simple harmonic motion resembles uniform circular motion when relating the two types of movement mathematically:

  • When viewing a marble moving along a circular ring from the side, it may appear similar to the one-dimensional back-and-forth motion of the ball on the spring.
  • If we define the radius of the ring to match the amplitude of the spring's oscillation and equate the marble's constant speed to the ball's maximum speed, they can be quantitatively analyzed using similar parameters:
    • The period of circular motion can also be described using amplitude and speed, yielding the same calculations as those derived in SHM.

Trigonometric Analysis of Position over Time

To illustrate how the ball’s position varies over time, we employ trigonometric analysis:

  1. The cosine of the angle (\theta) at any point along the marble's path corresponds with its horizontal distance from the ring’s center:
    \cos(\theta) = \frac{x}{A}
    This rearranges to determine the ball's position:
    x = A \cos(\omega t)

Upon graphing this equation, the result appears as a sinusoidal wave, reinforcing the relationship between SHM and waveforms.

The Role of Resonance in the Millennium Bridge

A crucial factor in the Millennium Bridge incident was resonance, which amplifies oscillations at specific frequencies. This is akin to how a child can swing higher through precise timing in pushing. When pedestrians leaned into the swaying, they inadvertently contributed to resonance, which intensified the oscillations in amplitude. Initially, the engineers accounted for vertical oscillations but did not consider the horizontal swaying induced by the crowd, leading to increased oscillatory motion and the eventual compound failure of the bridge.

Conclusion

In summary, today’s lesson provided a comprehensive introduction to simple harmonic motion, covering energy dynamics, maximum velocity equations, and the parallels with uniform circular motion. We also explored how the position of objects in simple harmonic motion evolves over time and the implications of resonance, illustrated through the Millennium Bridge incident. Further explorations on wave motion will be addressed in subsequent episodes of Crash Course Physics.