16.5 Energy and the Simple Harmonic Oscillator

Simple harmonic oscillators, like springs and pendulums, showcase fascinating energy transformations. As objects move back and forth, energy shifts between kinetic and potential forms, with the total energy remaining constant throughout the oscillation.

Understanding these energy changes helps us grasp the behavior of oscillating systems. By examining factors like spring stiffness, mass, and amplitude, we can predict an oscillator's maximum velocity and energy distribution, unlocking insights into everyday phenomena and engineering applications.

Energy in Simple Harmonic Oscillators

Maximum velocity in harmonic oscillators

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• Maximum velocity ($v_{max}$) reached at equilibrium position where displacement is zero
• Calculate $v_{max}$ using $v_{max} = \omega A$
  • $\omega$ represents angular frequency, calculated as $\omega = \sqrt{\frac{k}{m}}$
    • $k$ represents spring constant, a measure of spring stiffness (stiffer spring, higher $k$)
    • $m$ represents mass of the oscillating object (heavier object, lower $v_{max}$)
  • $A$ represents amplitude, maximum displacement from equilibrium (larger amplitude, higher $v_{max}$)
• Combine equations to get $v_{max} = A\sqrt{\frac{k}{m}}$
  • Example: A 0.5 kg mass attached to a spring with $k = 20 N/m$ oscillating with $A = 0.1 m$ has $v_{max} = 0.1\sqrt{\frac{20}{0.5}} = 0.632 m/s$

Energy transformations during oscillation

• Total energy remains constant in simple harmonic oscillator, sum of kinetic energy (KE) and potential energy (PE) always equals total energy
• Energy continuously transforms between KE and PE during oscillation
  • At equilibrium position:
    • KE reaches maximum, all energy is kinetic (object moving fastest)
    • PE is zero, no stored energy in spring or gravitational potential
  • At maximum displacement (amplitude):
    • KE is zero, object momentarily at rest
    • PE reaches maximum, all energy stored in spring or gravitational potential
• Maximum PE equals maximum KE, both equal to $\frac{1}{2}kA^2$
  • Example: A 0.2 kg mass on a spring with $k = 50 N/m$ oscillating with $A = 0.05 m$ has $PE_{max} = KE_{max} = \frac{1}{2}(50)(0.05)^2 = 0.0625 J$

Comparing Simple Harmonic Oscillators

Velocity factors in springs vs pendulums

• Spring-mass systems:
  • $v_{max}$ depends on spring constant ($k$), mass ($m$), and amplitude ($A$)
  • Calculate using $v_{max} = A\sqrt{\frac{k}{m}}$
  • Higher $k$ or $A$, or lower $m$, results in higher $v_{max}$
    • Example: Doubling $k$ or $A$, or halving $m$, increases $v_{max}$ by a factor of $\sqrt{2}$
• Pendulums:
  • $v_{max}$ depends on pendulum length ($L$), gravitational acceleration ($g$), and amplitude ($A$)
  • For small angles, calculate using $v_{max} = A\sqrt{\frac{g}{L}}$
  • Higher $A$ or $g$, or lower $L$, results in higher $v_{max}$
    • Example: Doubling $A$, using pendulum on Jupiter ($g = 24.79 m/s^2$), or halving $L$, increases $v_{max}$
• Similarities:
  • In both systems, $v_{max}$ is directly proportional to $A$
  • Factors affecting restoring force ($k$ in springs, $g$ and $L$ in pendulums) influence $v_{max}$

Characteristics of Simple Harmonic Motion

• Period: The time required for one complete oscillation
• Displacement: The distance of an object from its equilibrium position at any given time
• Oscillation: The repetitive back-and-forth motion of an object about its equilibrium position
• Resonance: The tendency of a system to oscillate with greater amplitude at certain frequencies
• Damping: The gradual reduction in the amplitude of oscillations due to energy dissipation