A-level Mathematics/OCR/M3/SHM

From Wikibooks, open books for an open world
< A-level Mathematics‎ | OCR‎ | M3
Jump to navigation Jump to search

SHM Equations of Motion[edit | edit source]

Defining Equation of Motion (in terms of acceleration and displacement)[edit | edit source]

By definition, a particle performs simple harmonic motion (SHM) when

  • the magnitude of its acceleration is proportional to the magnitude of its displacement from some fixed point, and
  • its acceleration and displacement are in opposite directions.

More formally, SHM occurs whenever the equation of motion holds, where is a positive constant. By convention, we write (we will explain the significance of later) so the equation becomes .

Simple harmonic oscillators are easy to find in the real world. The undamped oscillation of a mass attached to an elastic string or spring is just one example in the realm of mechanics. Before studying such examples, let us first examine the characteristics of SHM using the abstract model described below.

Consider the following oscillator with an oscillation amplitude of . Let denote its displacement from the origin, and its acceleration.

Shm1.png

Let us restate the defining equation for SHM:

Recall that the algebraic variables representing displacement and its derivatives wrt time represent one-dimensional vector quantities (i.e. they all lie on a line). If the value of the variable is positive, then its vector points in the positive direction; if it is negative, its vector points in the other direction.

Observe that when the , then . By Newton's Second Law, the resultant force acting on the oscillator is zero at this instance.

When (i.e. to the right of the origin), the resultant force points to the left, towards the origin. When , the resultant force points to the right, towards the origin. Because the resultant force always points towards the origin whenever the oscillator is not at the origin, the resultant force can be termed a restoring force. The behaviour of the restoring force suggests that the speed of the oscillator is maximum at the origin and is zero at the two ends where it changes direction (i.e. when ).

Equation of Motion, in terms of velocity, amplitude and displacement[edit | edit source]

Now, let us rewrite the defining SHM equation in terms of the velocity and displacement . We make the following substitution: . It follows that

(separating variables)

Using and as one pair of limits for the integral, we get

Now, it is clear that the speed is maximum when (in which case ), and when .

Equation of Motion, in terms of displacement, amplitude and time[edit | edit source]

The significance of the constant will become clear once we express the defining SHM equation in terms of the displacement and time :

This is just a second-order ordinary differential equation with constant coefficients. The characteristic equation (as a function of ) is given by:

Thus, the general solution to the differential equation may be written as:

, or equivalently, .

To solve for the pairs of constants or , we need a pair of initial conditions describing the SHM.

If the particle starts from rest at the maximum displacement , then we substitute and at to get , and , . The solution becomes .

On the other hand, if the oscillation begins at minimium displacement, then we obtain .

In general, we have

where the constant depends on the starting position of the oscillator.

These equations show that is just the constant angular velocity (or angular frequency) associated with the SHM. The following figure illustrates this relationship between SHM and uniform circular motion (with an angular velocity of ). The displacement of the simple harmonic oscillator is really just the -coordinate of the particle performing uniform circular motion. We can therefore think of SHM as a projection of uniform circular motion onto the diameter of the circle.

The simple harmonic oscillator completes one oscillation whenever it traverses twice the end-to-end distance (i.e. a distance of ). This corresponds to one revolution around the circle in the case of uniform circular motion. Thus, the period of oscillation in SHM is given by

Shm3.png

Example: Mass on an Elastic String[edit | edit source]

We wish to determine if a mass suspended on a light (i.e. massless) elastic string will demonstrate SHM.

Consider a light elastic string with natural length and modulus of elasticity suspended vertically from a fixed point O. The other end of the string is attached to a particle of mass .

Shm2.png

Let the point E denote the equilibrium position of the particle. At E, the particle is at rest (i.e. the resultant force acting on it is zero). Let denote the extension of the string when the particle is at E. Considering the vertical forces on the particle gives:

.

The displacement of the mass is always measured from the equilibrium position E, since this is the point where and hence, the resultant force .

Suppose the mass experiences a small displacement . Applying Newton's Second Law produces

(since )

which is the defining equation of motion for SHM, with . Note that the particle performs SHM only when the string remains taut, i.e. .

Summary[edit | edit source]

You are expected to know the following formulas, as they will not be provided in the examination:

Formula Description
SHM Defining Equation of Motion (in terms of acceleration and displacement)
SHM Equation of Motion, in terms of velocity, amplitude and displacement
SHM Equation of Motion, in terms of displacement, amplitude and time
SHM Period of Oscillation