15.2 Energy in Simple Harmonic Motion

Simple harmonic motion is all about energy transfer in oscillating systems. As objects bounce back and forth, energy constantly shifts between potential and kinetic forms, following predictable patterns based on position and velocity.

Understanding energy in these systems helps explain real-world oscillations, from playground swings to electrical circuits. We'll explore how to calculate energies, examine conservation principles, and see how forces relate to potential energy in equilibrium situations.

Energy in Simple Harmonic Motion

Energy calculations in harmonic oscillators

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• Potential energy ($PE$) in a simple harmonic oscillator depends on the spring constant ($k$) and the displacement from equilibrium ($x$) calculated using the formula $PE = \frac{1}{2}kx^2$
  • Spring constant ($k$) measures the stiffness of the spring (N/m)
  • Displacement ($x$) measures the distance from the equilibrium position (m)
• Kinetic energy ($KE$) in a simple harmonic oscillator depends on the mass ($m$) and the velocity ($v$) calculated using the formula $KE = \frac{1}{2}mv^2$
  • Mass ($m$) is the amount of matter in the oscillating object (kg)
  • Velocity ($v$) is the speed and direction of the oscillating object (m/s)
• Velocity in a simple harmonic oscillator can be calculated using the amplitude ($A$), angular frequency ($\omega$), and displacement ($x$) with the formula $v = \omega\sqrt{A^2 - x^2}$
  • Amplitude ($A$) is the maximum displacement from equilibrium (m)
  • Angular frequency ($\omega$) is the rate of oscillation (rad/s)
• Total energy ($E$) in a simple harmonic oscillator is the sum of potential and kinetic energy expressed as $E = PE + KE = \frac{1}{2}kx^2 + \frac{1}{2}mv^2$
  • Total energy remains constant throughout the oscillation (J)
  • This constant total energy is also known as the mechanical energy of the system

Conservation of energy in mass-spring systems

• In a simple harmonic oscillator, energy is continuously converted between potential and kinetic energy as the mass oscillates
  • At the equilibrium position ($x = 0$), potential energy is zero and kinetic energy is at its maximum (all energy is in the form of motion)
  • At the maximum displacement ($x = \pm A$), potential energy (elastic potential energy) is at its maximum and kinetic energy is zero (all energy is stored in the spring's compression or extension)
• Total energy remains constant throughout the oscillation, demonstrating the conservation of energy principle
  • As the mass moves from equilibrium to maximum displacement, potential energy increases while kinetic energy decreases (energy is transferred from motion to spring)
  • As the mass moves from maximum displacement to equilibrium, potential energy decreases while kinetic energy increases (energy is transferred from spring to motion)
• Conservation of energy in a simple harmonic oscillator can be expressed as $E = \frac{1}{2}kA^2 = \frac{1}{2}kx^2 + \frac{1}{2}mv^2$
  • Total energy ($E$) equals the maximum potential energy ($\frac{1}{2}kA^2$) and is always equal to the sum of instantaneous potential and kinetic energies

Force vs potential energy in equilibrium

• Force ($F$) in a simple harmonic oscillator is proportional to the displacement ($x$) and acts in the opposite direction, expressed as $F = -kx$
  • Negative sign indicates the restoring force acts opposite to the displacement
• Potential energy ($PE$) is related to the force by $PE = -\int F dx = \frac{1}{2}kx^2$
  • Potential energy is the work done against the restoring force to displace the mass
• Stable equilibrium occurs when the potential energy is at a minimum
  1. A small displacement from the stable equilibrium position results in a restoring force that pushes the mass back towards equilibrium
  2. The potential energy curve around a stable equilibrium point is concave up (U-shaped)
• Unstable equilibrium occurs when the potential energy is at a maximum
  1. A small displacement from the unstable equilibrium position results in a force that pushes the mass further away from equilibrium
  2. The potential energy curve around an unstable equilibrium point is concave down (inverted U-shaped)

Oscillations and Resonance

• An oscillation occurs when a system is displaced from its equilibrium position and experiences a restoring force
• The frequency at which a system naturally oscillates is called its natural frequency
• Resonance occurs when an external force drives a system at its natural frequency, resulting in a large increase in the amplitude of oscillation