Simple harmonic motion involves a continuous exchange of energy between kinetic and potential forms. As an object oscillates, its energy shifts back and forth, maintaining a constant total energy in ideal systems. This interplay is key to understanding oscillatory motion.
The total energy in simple harmonic motion is directly related to the amplitude of oscillation. Doubling the amplitude quadruples the energy, showcasing the importance of displacement in determining the system's overall energy state.
Energy in Simple Harmonic Motion
Kinetic and potential energy interchange
- Energy continuously converts between kinetic and potential forms maintaining constant total energy in ideal systems
- Maximum potential energy occurs at oscillation extremes with zero velocity and maximum displacement (stretched spring)
- Maximum kinetic energy at equilibrium position with maximum velocity and zero displacement (unstretched spring)
- Kinetic and potential energies vary sinusoidally out of phase by 90 degrees throughout cycle
Total energy calculation in SHM
- Total energy equation: $E = \frac{1}{2}kA^2$ where $E$ is total energy, $k$ is spring constant, $A$ is amplitude
- Energy components: Kinetic energy $KE = \frac{1}{2}mv^2$, Potential energy $PE = \frac{1}{2}kx^2$
- Energy conservation principle maintains constant total energy $E = KE + PE$
- Calculate using maximum displacement (amplitude) or maximum velocity
Amplitude and energy relationship
- Energy proportional to square of amplitude doubling amplitude quadruples energy
- Larger amplitude increases maximum potential and kinetic energies indicating more energy in system
- Energy-amplitude graph shows parabolic shape
Energy conservation in oscillations
- Damped oscillations gradually decrease total energy due to friction or air resistance with exponentially decreasing amplitude
- Driven oscillations receive energy from external force resonating when driving frequency matches natural frequency
- Steady-state driven oscillations balance energy input from driving force with energy dissipated by damping
- Oscillation types: Undamped (constant total energy), Damped (decreasing total energy), Driven (variable total energy)