S6.2 Simple Harmonic Motion: the physical context
[A] Kinds of equilibrium
- A body subject to no net external force or torque, and initially at rest, will remain so (Key Point 2.1): it is in static equilibrium.
- We can usefully distinguish between three kinds of static
equilibrium which differ according to the forces called into play
when the body is displaced a little from the equilibrium position.
- In unstable equilibrium (figure (a)) the forces drive the body further away from the equilibrium position.
- In stable equilibrium (figure (b)) the forces drive the body back towards the equilibrium position.
- In neutral equilibrium (figure (c)) the body continues to experience no net force.
We are concerned entirely with systems near a point of stable equilibrium.
Why?
RevealHideWhy?
There are two reasons for focusing on this case. First there is not much to be said of a general character about the other two cases:
- What an unstable system does as it moves further and further away from equilibrium depends on what it finds ’out there’, and has to be dealt with on a case-by-case basis.
- A system in neutral equilibrium ’does’ nothing at all!
The other reason is that stable-equilibrium is arguably the ’norm’ in our world, where there are always likely to be disturbances coming from ’the rest of the universe’ that will drive an unstable system to seek out some other arrangement.
[B] What happens near stable equilibrium
- Consider the behaviour of a system near a position of stable equilibrium.
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- We can describe the behaviour in two entirely equivalent ways:
- in the language of forces
- in the language of potential energies
- The force called into play is linear and restoring
(S2.9):
F(x)=-kxwith k a positive constant.
- The associated potential energy is given by
where U(0) is a constant (frequently set to zero).
Puzzled?
You should recall that in defining a potential energy we have to ’fix’ a value at some point: thus when we say that a body at height h (above some surface) ’has’ a (gravitational) potential energy of mgh we are implicitly using the choice (assumption, convention …) that the PE is zero at the surface itself. - The two functions are related by Key Point 3.11:
The fact that x=0 is a point of equilibrium implies that the force is zero there. Hence
so U(x) has a turning point at x=0.
The fact that x=0 is a point of stable equilibrium implies that the force is restoring ie that k>0. Hence
implying that the turning point is a minimum.
- These results hold true more generally:
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![\[ F(x) = -kx \hspace{0.2cm}\mbox{\rm and}\hspace{0.2cm} U(x) = U(0) + \frac{1}{2}kx^2 \]](mastermathpng-4.png)

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
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
- If x=0 is a point of equilbrium F(x=0) must vanish so
Then
and- For x=0 to be stable, k must be positive, implying that x=0 is a minimum of U(x).
[C] SHM: the key equation
- Again consider the specific case of the mass-spring system.
- Suppose that (at some instant) the mass, m say, has some displacement x.
- Its acceleration at that instant follows from Newton’s 2nd Law
(Key Point 2.2)
where we have assumed that the only force the mass experiences is that due to the spring.
We will write this equation in the form
where
At this point ω is a convenient abbreviation; note that its units are s-1.
- Equation 6.3 is the equation of motion for the mass: it expresses the acceleration at an instant in terms of the displacement at that instant.
- The motion that emerges from this equation (to be explored below) is known as simple harmonic motion (SHM).
- We may generalise as follows:
Key Point 6.2
The equation of motion of a system near to a point of stable equilibrium is of the form![\[ \ddot{x} = -\omega^2 x \]](mastermathpng-14.png)

Commentary
- This equation is ubiquitous –it features in virtually every branch of physics (and beyond).
The variable x need not represent a displacement
- 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
![]() | HRW section 15.2-15.3 |
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