Energía cinética, potencial y mecánica en un oscilador armónico simple
Energy of a Simple Harmonic Oscillator
Introduction to Simple Harmonic Motion
- The tutorial focuses on the energy associated with a simple harmonic oscillator, specifically in the context of oscillatory motion.
- It introduces a mass-spring system where total mechanical energy is the sum of kinetic energy and potential elastic energy.
Total Mechanical Energy Conservation
- In the absence of frictional forces, total mechanical energy remains constant throughout the motion.
- Kinetic energy (E_c) is defined as 1/2 m v^2 , where velocity (v) is expressed as -omega A sin(omega t + phi) .
Kinetic and Potential Energy Formulas
- The expression for kinetic energy becomes E_c = 1/2 m omega^2 A^2 sin^2(omega t + phi) .
- Potential elastic energy (E_p), stored in the spring during elongation x, is given by E_p = 1/2 k x^2 .
Displacement and Energy Relationships
- Displacement for simple harmonic motion is defined as x = A cos(omega t + phi) , leading to potential energy being expressed as E_p = 1/2 k A^2 cos^2(omega t + phi) .
- The total mechanical energy equation combines both energies: E_total = E_c + E_p.
Deriving Total Mechanical Energy Expression
- Substituting equations for kinetic and potential energies into the total mechanical energy equation reveals that it consists of sine and cosine terms.
- Angular frequency ( ω) relates to spring constant (k/m), allowing simplification of expressions.
General Case Considerations
- The general case considers phase angle ( φ), which can vary; if released from rest at maximum displacement, then φ = 0.
- Factoring out common terms leads to an expression for total mechanical energy:
- E_total = ½ k A^2 (sin^2(omega t + φ)+cos^2(omega t + φ)).
Conclusion on Energy Characteristics
- Notably, since sine squared plus cosine squared equals one, this simplifies to:
- E_total = ½ k A^2.
Energy in Simple Harmonic Motion
Understanding Energy Distribution in Oscillators
- The total mechanical energy of a harmonic oscillator is equal to the potential elastic energy when the displacement x equals the amplitude, indicating that kinetic energy is zero at this point.
- At equilibrium position ( x = 0 ), all mechanical energy converts into kinetic energy, which reaches its maximum value while potential energy becomes zero.
- The relationship between kinetic and potential energies can be expressed as: Total Mechanical Energy = Kinetic Energy + Potential Energy. This holds true for any position x .
- For positions other than amplitude or equilibrium, one must apply the equation for total mechanical energy, which remains constant throughout the motion.
Graphical Representation of Energies
- Two important graphs illustrate how kinetic and potential energies vary over time; both remain positive and their sum equals total mechanical energy, represented as 1/2 k A^2 .
- Continuous transformation occurs between kinetic and potential energies; at maximum displacement (amplitude), all energy is stored as potential, while at equilibrium, it is entirely kinetic.
Example Calculation of Total Energy
- An example involves a 1.0 kg mass attached to a spring with a spring constant of 40.0 N/m oscillating on a frictionless surface; we need to calculate its total mechanical energy given an amplitude of 0.06 m.
- The absence of friction ensures no loss of mechanical energy during oscillation; thus, total mechanical energy remains consistent across all points in motion.
- Using the formula for total mechanical energy: E_total = 1/2 k A^2 , where k = 40.0 N/m and A = 0.06 m .
- Substituting values yields:
- E_total = 1/2 (40.0 N/m)(0.06 m)^2 = 0.072 J.