6.2 Energy of a Simple Harmonic Oscillator

6.2 Energy of a Simple Harmonic Oscillator

Conservation of Energy in SHM

Energy in a simple harmonic oscillator is conserved. This means that the total energy remains constant as it oscillates between potential and kinetic energy.

Internal Energy

A system with internal structure can have internal energy, which can change due to the system’s internal structural changes.

Key Points:

• Definition: Internal energy (ΣU) refers to the random, chaotic motion of an object’s particles and is a measure of thermal energy.

• SHM Context: Internal energy manifests as elastic potential energy when displaced from equilibrium. This energy alternates between kinetic and potential energy during oscillation.

• Behavior: The internal energy in SHM is periodic. It’s maximum at maximum displacement and minimum at equilibrium.

• Equation:

  where is the spring constant, and is the displacement.

• Significance: Internal energy helps explain energy conversions in SHM.

Potential Energy in SHM

Potential energy exists within systems when objects interact with conservative forces.

Key Points:

• Definition: Potential energy (ΣU) is stored energy based on position or configuration.

• SHM Context: In oscillators, potential energy is stored elastically due to spring deformation when displaced.

• Behavior: Periodic, maximum at maximum displacement, and minimum at equilibrium.

• Equation:

• Significance: Understanding potential energy is crucial for analyzing energy storage and conversion.

Kinetic Energy in SHM

Kinetic energy is associated with the motion of objects.

Key Points:

• Definition: Kinetic energy (ΣK) quantifies the ability to do work due to motion.

• SHM Context: Maximum at maximum velocity (equilibrium), and minimum at maximum displacement.

• Behavior: Periodic, alternating with potential energy.

• Equation:

  where is the mass, and is velocity.

• Significance: Kinetic energy reveals motion dynamics during oscillation.

Energy Conversion in SHM

The total energy in SHM alternates between potential and kinetic energy, maintaining a constant sum.

Key Features:

• At Maximum Displacement: Potential energy is maximum, kinetic energy is zero.

• At Equilibrium: Kinetic energy is maximum, potential energy is zero.

• Graph Insights:

  • Total energy remains constant.

  • Potential and kinetic energy graphs follow a periodic curve due to the squared terms in their equations.

  • Peaks in potential energy align with displacement maxima, while kinetic energy peaks coincide with velocity maxima.

Example Problems

Example 1: Total Energy at Maximum Displacement

Problem: A 1 kg mass on a spring () oscillates vertically without friction. The displacement is 0.2 m from equilibrium. Calculate the total energy.

Solution:

• Total energy is the sum of potential and kinetic energy.

• At maximum displacement, velocity is zero, so energy is entirely potential.

Answer: Total energy is .

Example 2: Total Energy at Equilibrium

Problem: A 2 kg mass on a spring () oscillates vertically without friction. The displacement is 0.5 m from equilibrium. Calculate the total energy at equilibrium.

Solution:

• At equilibrium, potential energy is minimum (zero), and total energy equals kinetic energy.

Answer: Total energy is .