Question

Find the equation of motion of a particle moving along the \(x\) axis if the potential energy is \(V=\frac{1}{2} k x^{2} .\) (This is a simple harmonic oscillator.)

Short answer

The equation of motion is \( x(t) = A\cos \left( \sqrt{\frac{k}{m}} t + \phi \right) \).

Step-by-step solution

Identify the Potential Energy and Force

The potential energy given is \[V=\frac{1}{2} k x^{2}\]. Identify the force acting on the particle using the relation \[ F = -\frac{dV}{dx} \].

Compute the Force

Differentiate the potential energy with respect to x:\[ F = -\frac{d}{dx} \left( \frac{1}{2} k x^{2} \right) = -k x \].

Apply Newton's Second Law

Newton's second law states that \[ F = ma \] where \(a\) is the acceleration. Substitute the expression for force to get \[ -kx = m\frac{d^{2}x}{dt^{2}} \].

Derive the Equation of Motion

Rearrange the equation to obtain the standard form of the equation of motion for simple harmonic motion:\[ \frac{d^{2}x}{dt^{2}} + \frac{k}{m}x = 0 \].

Write the Solution for the Equation of Motion

The general solution for the equation of motion \( \frac{d^{2}x}{dt^{2}} + \omega^{2}x = 0 \) where \( \omega = \sqrt{\frac{k}{m}} \), is \[ x(t) = A\cos(\omega t + \phi) \].

Key concepts

Simple Harmonic Oscillator

A simple harmonic oscillator is a system that experiences a restoring force proportional to its displacement from equilibrium.
Imagine you've attached a mass to a spring. When you pull it and then release, the mass vibrates back and forth. This is simple harmonic motion.
Based on the exercise, we know the potential energy is given by \[ V = \frac{1}{2} k x^{2} \]. Here, \(k\) is the force constant and \(x\) is the displacement.

Newton's Second Law

Newton's second law is all about understanding how forces influence motion.
It states that the force acting on an object is equal to the mass of the object multiplied by its acceleration: \[ F = ma \].
In our problem, by combining this with the force derived from the potential energy, we find \[-kx = m\frac{d^{2}x}{dt^{2}} \]. This is the foundational principle that allows us to describe the motion of the oscillator.

Potential Energy

Potential energy represents the stored energy of an object due to its position or state.
In our scenario, the given potential energy is \[ V = \frac{1}{2} k x^{2} \]. This form indicates we are dealing with a spring-like system.
When we differentiate this potential energy with respect to \(x\), we get the force: \[ F = - \frac{dV}{dx} = -kx \], which is the classic restoring force for a simple harmonic oscillator.