Question

A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by ?

Answer 1

Explanation:

This is a simple harmonic motion, produced by the action of a driving force that is directly proportional to the position of the body. Its position is described as a function of time by a sinusoidal function. According to Newton's second law and Hooke's law:

$F=m\frac{d^2x}{dt^2}\\F=-kx$

Here m is the object's mass, k is the spring constant and x is the displacement from the equilibrium position.

$m\frac{d^2x}{dt^2}=-kx\\\frac{d^2x}{dt^2}=-\omega_0^2x\\x=Asin(\omega_0 t-\phi)$

With $\omega_0=\sqrt{\frac{k}{m}}$, called natural frequency, A is the amplitude, that is the maximum displacement for the equilibrium position and $\phi$ is the initial phase.