Periodic Motion: Simple Harmonic Motion, Pendulums, Damping, and Resonance

Periodic Motion and Oscillations

Introduction to Simple Harmonic Motion (SHM)

Simple Harmonic Motion (SHM) is a type of periodic motion where an object moves back and forth about an equilibrium position under a restoring force proportional to its displacement. SHM is fundamental in physics, describing systems such as springs, pendulums, and molecular vibrations.

• Definition: SHM occurs when the restoring force is directly proportional to the displacement and acts in the opposite direction: $F = -kx$ (Hooke's Law).

• Mathematical Form: The position as a function of time is sinusoidal: $x(t) = A \\cos(\omega t + \phi)$, where $A$ is amplitude, $\omega$ is angular frequency, and $\phi$ is phase.

• Velocity and Acceleration:

  • Velocity: $v(t) = -A\omega \\sin(\omega t + \phi)$

  • Acceleration: $a(t) = -A\omega^2 \\cos(\omega t + \phi)$

• Relation between Position, Velocity, and Acceleration: All are sinusoidal functions of time, but with phase differences.

• Time Period and Frequency:

  • Angular frequency: $\omega = \sqrt{\frac{k}{m}}$

  • Time period: $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}}$

  • Frequency: $f = \frac{1}{T}$

• Example: A glider of mass $m = 0.50$ kg attached to a spring with $k = 200$ N/m exhibits SHM. To double the frequency, $k$ must be increased by a factor of 4 (since $f \propto \sqrt{k}$).

Pendulums

Pendulums are classic examples of systems exhibiting periodic motion. Their oscillations are governed by gravitational restoring forces and depend on the length of the pendulum, not its mass.

• Simple Pendulum: Consists of a mass (bob) attached to a string of length $l$.

• Restoring Force: For small angles ($\theta$), the restoring force is $F = -mg\sin\theta \approx -mg\theta$.

• Time Period: $T = 2\pi \sqrt{\frac{l}{g}}$

• Frequency: $f = \frac{1}{2\pi} \sqrt{\frac{g}{l}}$

• Effect of Gravity: On planets with different gravitational acceleration (e.g., Jupiter or the Moon), the period changes accordingly.

• Example: A pendulum on the Moon (lower $g$) will have a longer period than on Earth.

Foucault Pendulum

The Foucault pendulum is a large, freely swinging pendulum that demonstrates Earth's rotation. Its plane of oscillation rotates over time due to the Coriolis effect.

• Application: Used to provide experimental evidence of Earth's rotation.

• Historical Note: Named after Leon Foucault, who first demonstrated this effect in 1851.

Driven Oscillations

Driven oscillations occur when an external force periodically drives a system. The response depends on the frequency of the driving force relative to the system's natural frequency.

• Natural Frequency: The frequency at which a system oscillates when not driven by an external force.

• Resonance: Occurs when the driving frequency matches the natural frequency, resulting in large amplitude oscillations.

• Example: Pushing a swing at its natural frequency increases its amplitude efficiently.

Damped Oscillations

Damping refers to the gradual loss of energy in an oscillating system, usually due to friction or resistance, causing the amplitude to decrease over time.

• Types of Damping:

  • Light Damping: Oscillations persist but gradually die out.

  • Heavy Damping: Oscillations die out quickly.

• Mathematical Form: $x(t) = Ae^{-\gamma t} \\cos(\omega' t + \phi)$, where $\gamma$ is the damping coefficient.

• Example: A mass-spring system in oil exhibits damped oscillations.

Driven Damped Oscillators and Resonance

When a damped oscillator is driven by an external force, its amplitude depends on the driving frequency and the amount of damping present.

• Resonance Peak: Lightly damped systems show a sharp resonance peak; strong damping reduces the peak and broadens the response.

• Practical Implications: Resonance can cause structural failures (e.g., bridges) if not properly managed.

• Example: The collapse of the Tacoma Narrows Bridge was due to resonance effects.

Summary Table: Key Properties of Oscillatory Systems

System	Restoring Force	Time Period ($T$)	Depends on
Mass-Spring (SHM)	$F = -kx$	$2\pi \sqrt{\frac{m}{k}}$	Mass $m$, Spring constant $k$
Simple Pendulum	$F = -mg\sin\theta$	$2\pi \sqrt{\frac{l}{g}}$	Length $l$, Gravity $g$
Damped Oscillator	$F = -kx - b v$	Longer than undamped $T$	Damping coefficient $b$
Driven Oscillator	External periodic force	Depends on driving frequency	Driving frequency, damping

Additional info:

• For small angles ($\theta$), $\sin\theta \approx \theta$ (in radians), which allows the pendulum to be modeled as SHM.

• Resonance is a critical concept in engineering, requiring careful design to avoid destructive oscillations.