Question

The total energy of simple harmonic oscillations is directly proportional to?

Answer 1

The total energy of a simple harmonic oscillator is directly proportional to the square of its amplitude.

A simple harmonic oscillator is a system that exhibits periodic motion around a fixed equilibrium point, where the motion is sinusoidal and can be described by the equation:

$x(t) = \text{A sin(ωt + φ)}$

where x is the displacement of the oscillator from its equilibrium position, A is the amplitude of the motion, ω is the angular frequency of the oscillation, t is time, and φ is the phase angle.

The total energy of the oscillator is the sum of its kinetic energy and potential energy, and it is given by the equation:

$E = (\frac{1}{2}) k A^2$

where k is the spring constant of the oscillator. As we can see from this equation, the total energy of the oscillator is directly proportional to the square of its amplitude. This means that as the amplitude of the oscillator increases, so does its total energy.