9.1 Simple harmonic motion
Simple harmonic motion is motion where the acceleration is proportional to displacement, that is
a = -ω2x,
where ω is the angular frequency, and x is the displacement. Since acceleration is the second derivative of displacement, we can solve the differential equation (not in the IB syllabus) and find the equations of motion. The displacement is therefore
x = x0 sin ωt,
where we have chose t in such a way that the displacement at t=0 is zero. We differentiate this once to get velocity,
v = ωx0 cos ωt.
Now that we have the velocity, we can calculate the kinetic energy of a system undergoing simple harmonic motion,
Ek = 1/2 m v2
= 1/2 m (ωx0 sin ωt)2
= 1/2 m ω2 x02 cos2 ωt.
Let’s now consider a mass on a table, attached to a wall with a spring with spring constant K. When displaced, the net force acting on it is
F = -Kx.
We also know that the acceleration is proportional to its displacement, hence
ω2 = K/m.
Hence, the potential energy is thus
Ep = 1/2 Kx2
= 1/2 (ω2 m) (x0 sin ωt)2
= 1/2 m ω2 x02 sin2 ωt.
The total energy is
E = Ek +Ep
= m ω2 x02 ,
where we have used sin2 ωt + cos2 ωt = 1, a trigonometric identity. The total energy of the system does not change with time, which is what we would expect from the conservation of energy. Of course, this is not true if the system loses energy to the surroundings (a damped system), but that is not discussed here.
9.2 Single slit diffraction
A powerful tool one can use to understand the behaviour of waves is the Huygens principle. It states that every point on a wavefront is a point source. The interference from every source interfering with every other source gives the subsequent wavefront.
With this in mind, what happens when a plane wave passes through a narrow slit? Since the formerly-infinite wavefront now is limited in spatial extent, it behaves differently at the edges. When it was a plane wave, every source had other sources on the left and right, which is why the wavefront continued to remain straight. However, the sources on either edge of the slit no longer have other sources on one side to balance them out. Hence, the wave spreads out. This is diffraction.
The diffraction pattern from a single slit is complicated, having many different maxima in minima. The key thing to note is that there is a single maximum in the middle. The location of the minima is given by
nλ = d sin θn.
Here n is an integer (0, 1, 2, 3, …), and theta is the angle between the line perpendicular to the slit and the line joining the slit to the minimum.
(Diagrams for reflection and single slit diffraction)
When two waves superpose, the resultant wave has a different amplitude from the two original waves. When the two waves add, we say that there is constructive interference. When they subtract, we say that the interference is destructive.
Near the central maximum, the fringe separation x is
x = λL / d,
where L is the perpendicular distance between the center of the slits and the screen, and d is the slit separation. This is a small angle approximation of the more general formula for a diffraction grating. It arises because the slits have a finite width, and we know from single slit diffraction, the resulting envelope has a finite extent in space. Hence, far from the central maximum (large θ), there is no intensity from either slit, hence there can be no interference pattern to observe.
We now consider a diffraction grating, a large array of tiny slits, where the slits are spaced a distance d apart from their neighbours. The locations of the maxima are given by
mλ = a sin θm.
Where m is an integer and a is the size of the slit.
Thin film interference (qualitatively).
θ = 1.22 λ / a
Resolving two sources
Diffraction gratings and resolution
9.5 Doppler effect
The equation and knowing how to use it