Energy In Simple Harmonic Motion (Copy)

Total Mechanical Energy in SHM

• In ideal simple harmonic motion, total mechanical energy is conserved
• This total energy is the sum of kinetic energy (KE) and potential energy (PE) at any point in the oscillation
• Energy continually transforms between KE and PE during the motion, but the total remains constant
• Total energy equation:
  • E = ½mω²x₀²
    • E = total mechanical energy
    • m = mass of oscillating object
    • ω = angular frequency
    • x₀ = amplitude of oscillation
• This energy is proportional to the square of the amplitude (x₀²)

Kinetic Energy in SHM

• Kinetic energy is highest at the equilibrium position (where displacement x = 0)
• It is lowest (zero) at the maximum displacement (x = ±x₀)
• Kinetic energy expression:
  • KE = ½mv²
  • Since v = ±ω√(x₀² − x²) in SHM:
    • KE = ½mω²(x₀² − x²)
• Key facts:
  • KE is maximum at equilibrium
  • KE is zero at turning points (x = ±x₀)

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Potential Energy in SHM

• PE in SHM refers to elastic potential energy, gravitational potential energy, or restoring force work, depending on the context
• PE is maximum at the maximum displacement (x = ±x₀)
• PE is zero at the equilibrium position
• Expression for potential energy:
  • PE = ½mω²x²
    • x = instantaneous displacement from equilibrium
• At any point:
  • Total Energy E = KE + PE
  • ½mω²x₀² = ½mω²x² + ½mω²(x₀² − x²)

Energy Interchange Summary

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Graphical Representation of Energy in SHM

• Kinetic Energy vs Time:
  • Sinusoidal graph
  • Maximum at equilibrium
  • Zero at x = ±x₀
  • Frequency = 2f (twice the oscillation frequency)
• Potential Energy vs Time:
  • Also sinusoidal
  • Zero at equilibrium
  • Maximum at x = ±x₀
  • Also has frequency = 2f
• Total Energy vs Time:
  • Constant horizontal line
  • Indicates no energy loss in an ideal SHM system

Graphical Representation of Energy vs Displacement

• KE vs Displacement:
  • Inverted parabola
  • Maximum at x = 0
  • Zero at x = ±x₀
• PE vs Displacement:
  • Parabolic curve
  • Zero at x = 0
  • Maximum at x = ±x₀
• E_total:
  • Straight horizontal line at E = ½mω²x₀²
  • Does not change with x

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Key Mathematical Derivations

Total Energy:

• Derived by setting:
  • E = KE_max = ½mω²x₀²

KE in terms of x:

• From v = ω√(x₀² − x²):
  • KE = ½m(ω²)(x₀² − x²)

PE in terms of x:

• PE = ½mω²x²
• Total energy:
  • E = KE + PE = ½mω²(x₀² − x²) + ½mω²x² = ½mω²x₀²

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Important Observations

• Energy transformation in SHM is continuous and periodic
• The rate of energy transformation varies with position:
  • Fastest transformation near equilibrium (greatest velocity)
  • Slowest near turning points (zero velocity)
• If damping is introduced:
  • Total energy decreases over time
  • KE and PE still interchange, but the maximum values decrease
  • Energy is lost to the surroundings, typically as heat

Units Recap