Oscillatory Motion: Resonance

Natural Frequency

  • Every object has a natural frequency, which is the rate at which it oscillates when undisturbed.

  • Examples:

    • Pendulum: Swings at its natural frequency when released.

    • Mass-spring system: Oscillates at its natural frequency.

    • Torsion pendulum: Twists at its natural frequency.

  • To change the natural frequency, you must alter the factors that affect it:

    • Pendulum: Change the length of the arm or gravity.

    • Mass-Spring: Change the stiffness of the spring or mass.

    • Torsion pendulum: Change the stiffness of the wire or the mass distribution along the spinning bar.

  • A tuning fork rings at a constant frequency unless disturbed.

  • Changing the frequency of a complex object like a tuning fork involves altering its structure, including:

    • Shape

    • Length of arms

    • Width of arms

    • Metal used

    • Distance between the forks

    • Shape of the neck

  • Tuning a tuning fork can be achieved by adding weights along the length of the forks.

Phase

  • The oscillating motion can be represented graphically as a sine wave, plotting amplitude or displacement against time.

  • Multiple sine waves can be superimposed.

  • Sine waves are useful due to the mathematical tools available to describe them.

  • Phase Difference: Describes the relative positions of two objects in motion along their cycles.

    • In Phase: When their positions line up.

    • Opposite Phase: When they are at opposite positions.

    • Out of Phase: Any other relative position that is not in phase or opposite phase.

Resonance

  • Most oscillations are driven oscillations, requiring a force to maintain the cycle.

  • Drive Oscillations: Driven oscillations, also known as forced oscillations, occur when a system oscillates due to an external, periodic driving force. Unlike free oscillations, where the system oscillates on its own after an initial disturbance, driven oscillations are maintained by the continuous application of the external force. 

  • Resonance: Occurs when the driving oscillations are in phase with the object's natural frequency.

    • Pushing someone on a swing: Pushing in phase achieves the highest amplitude with the least force. Pushing out of phase cancels out the motion.

  • All objects, including buildings, have a natural frequency.

  • 1985 Mexico City Earthquake: Damage was uneven because the frequency of the earthquake matched the natural frequency of some buildings, leading to large amplitude waves and structural failure.

  • Engineers must consider the potential for resonance in structures during design.

  • Tacoma Narrows Bridge (1940): The wind's frequency matched the bridge's natural frequency, causing large amplitude waves and eventual collapse.

  • Determining the natural frequency of complex objects can be challenging.

  • One method involves driving the object at different frequencies to identify the large amplitude build-up associated with resonance.

  • Example: Singing notes to a wine glass to find the frequency at which it breaks, which corresponds to its natural frequency.

  • Connecting multiple pendulums together results in them lining up over time.