Hidden Content Control Panel: Reveal All | Hide All

S6.2 Simple Harmonic Motion: the physical context

[A] Kinds of equilibrium

[B] What happens near stable equilibrium

  • To be specific, we will choose the mass-spring system (figure (b) above; and right)
  • But we will then see that our arguments are general.
Graphic - No title - shmaspring

Key Point 6.1

The behaviour of a body given a small displacement from a point of stable equilibrium can be described, equivalently, by a force F(x) and a potential energy U(x) with the forms
\[ F(x) = -kx \hspace{0.2cm}\mbox{\rm and}\hspace{0.2cm} U(x) = U(0) + \frac{1}{2}kx^2 \]
with k>0. The point x=0 is a minimum of the potential energy.

Commentary

Commentary

  • Let us explore how these claims emerge in such a general form.
  • Consider a system subject to some arbitrary force F(x) and associated potential U(x).
  • Consider the behaviour of U(x) near x=0.
  • According to Taylor’s theorem, for any reasonably physical function, for small x one can write the approximation

    \[ U(x) = \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{ U(0) + x \frac{dU}{dx}\mid _{x=0} + \frac{x^2}{2} \frac{d^2U}{dx^2}\mid _{x=0} + \ldots} \]

    where …stands for terms proportional to x3 or still higher powers of x.

  • The approximation gets better and better the smaller is x.
  • The implied force is then
    \[ F(x) = - \frac{dU(x)}{dx} = \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{ -\frac{dU}{dx}\mid _{x=0} - x \frac{d^2U}{dx^2}\mid _{x=0} + \ldots } \]
  • If x=0 is a point of equilbrium F(x=0) must vanish so
    \[ \frac{dU}{dx}\mid_{x=0}= 0 \]
  • Then

    \[ U(x) = \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{U(0) + \frac{1}{2}kx^2 + \ldots} \]
    and
    \[ F(x) = \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{-x \frac{d^2U}{dx^2}\mid _{x=0} + \ldots = -kx +\ldots } \]

    Expanding the maths.

  • For x=0 to be stable, k must be positive, implying that x=0 is a minimum of U(x).

About turning points

About turning points

The fact that the first derivative of U(x) vanishes at x=0 tells us that x=0 is a turning point. The fact that the second derivative is positive tells us that this turning point is a minimum
 
 

[C] SHM: the key equation

Key Point 6.2

The equation of motion of a system near to a point of stable equilibrium is of the form
\[ \ddot{x} = -\omega^2 x \]
where ω is a constant, characteristic of the system. This is the fundamental equation of simple harmonic motion. A system obeying this equation is described as a simple harmonic oscillator.

Commentary

Commentary

  • This equation is ubiquitous –it features in virtually every branch of physics (and beyond).
  • The variable x need not represent a displacement

    Think about units?

    Think about units?

    Note that the units of x appear on both sides of the SHM equation so our conclusion about the units of ω holds true irrespective of what x happens to represent.
     

  • Applications of the SHM equation exist in which x represents:
    • an angle (in a pendulum) measured in radians
    • a current (in a circuit) measured in amps
    • an electric field (in a laser cavity), measured in volts/metre
  • We will first establish physically and mathematically the general properties of SHM that follow from this equation.
  • We shall then apply these general results to specific examples of systems near statistical equilibrium
 

Learning Resources

Textbook: HRW section 15.2-15.3
Course Questions:
Self-Test Questions: