Simple Harmonic Motion: Kinetic and Potential Energy Analysis

Simple Harmonic Motion: Energy Analysis

The energy of a particle performing simple harmonic motion is composed of two contributions: the kinetic energy E_c, associated with the particle's velocity, and the potential energy E_p, due to the restoring force. The displacement of the movement is described by the expression x = A sin (ωt + φ), speed is v = dx / dt = Aω cos (ωt + φ), and the acting force (F = -Kx) is associated with a potential energy of elastic type: E_p = ½ kx².

Potential and Kinetic Energy Equations

Thus, the potential energy is E_p = ½ kA²sin²(ωt + φ), and the kinetic energy is: E_c = ½ mv² = ½ mA²ω²cos²(ωt + φ) = ½ kA²cos²(ωt + φ) where k = mω²

Total Energy in Simple Harmonic Motion

Therefore, the total energy is: E_t = E_c + E_p = ½ kA²cos²(ωt + φ) + ½ kA²sin²(ωt + φ) = ½ kA² = ½ mv²_max. The total energy in simple harmonic motion remains constant. It is equal to the maximum kinetic energy and equal to the maximum potential energy. There is a continuous transformation of kinetic energy into potential energy, and vice versa.

Huygens' Principle and Wavefronts

This is a simple mechanism for the construction of wavefronts from previous fronts. A wavefront is one of the surfaces that pass through the points where a wave oscillates with the same phase. Huygens' principle states that: The points in a wavefront become sources of secondary waves, whose envelope is a new primary wavefront.

Applying Huygens' Principle

The way to apply it is: plot small circles of the same radius with centers at different points on a wavefront, and then plot the envelope of the circles, which is the new wavefront. The figure shows an example of application to a spherical wavefront and another example to explain the diffraction of a plane wave against an obstacle.

Consequences of Huygens' Principle

One consequence of Huygens' principle is that all the rays take the same time between two consecutive wavefronts. Rays are lines perpendicular to the wavefronts and correspond to the line of wave propagation. Although Huygens' principle was formulated for matter waves, which were the only ones known at the time, its principle is valid for all types of waves. Kirchhoff extended the method to electromagnetic waves once they were discovered.