Related chapter: Waves
- 9.1 Simple harmonic motion
- State the defining equation for SHM and associate it with the graph of against .
- Understand that refers to the displacement from the equilibrium position.
- Associate with the hardware of the system generating the SHM.
- From calculate and , understanding that they do not change with , the amplitude.
- Identify (in the data booklet) and apply the equations relating to:
- as a function of .
- as a function of .
- as a function of .
- kinetic energy as a function of .
- total energy as a function of amplitude.
- Sketch the graphs for the relationships above.
- Show graphically how the kinetic and potential energy changes with displacement and with time.
- Discuss the parameters of a mass–spring system that affect its frequency and apply the correct equation.
- Discuss the parameters of a pendulum that affect its frequency and apply the correct equation.
||What it represents
||length of pendulum
||acceleration due to gravity
Single slit diffraction
First location of destructive interference:
For small angles of :
Double slit diffraction
Separation between maximums:
Multiple slit diffraction
With slits, there are maxima.
Intensity of the central maximum: , where is the intensity of the central maximum in single slit diffraction.
- Simple harmonic motion can be modelled by the following equations:
- : amplitude
- : angular frequency (how many radians on the graph correspond to 1 second)
- : the leftward phase shift
- : time
- : displacement from equilibrium position.
- : velocity.
- : acceleration.
- In addition:
Velocity and acceleration as a function of displacement
- In an ideal system with no drag forces, energy is conserved:
- The total energy of a system can be expressed in terms of the maximum kinetic energy of the system:
- So kinetic and potential energy are:
- note that and .
Single slit diffraction
- For destructive interference, the path difference must be , and the first minimum is obtained where path difference = .
- With the aid of the diagram, we know that AB is the path difference between the two waves, the slit width is , and is the wavelength:
- So, we can say that the first minimum for a single slit diffraction is observed at an angle where .
- The formula to find the angle at which additional minima form becomes:
- : slit width.
- : the nth minima.
- : wavelength.
- We can also conclude that:
- As wavelength , , and the angular width of the central maxima .
- As , , and so the angular width of the central maxima .
Young’s Double Slit Experiment
- When light from two slits intefere in the following setup, a set of fringes are formed:
- The following conditions are required:
- The light from both slits must be coherent, ie, constant or zero phase difference.
- The distance of between the slits and the slit width must be negligible compared to the distance between the screen and the slits.
- The slit width should be comparable to the wavelength of the light for circular diffraction.
- Since , it can be assumed that the two rays are parallel.
- For constructive interference at point , , where .
- In this case, .
- We can assume because is extremely small.
- Finally, we can conclude that:
The graph of light intensity vs angle from center for both single and double slit diffraction is as follows:
- The intensity of the double slit pattern is modulated by the one-slit pattern.
- If the number of slits are increased (with the same length between each slit), the fringes are more distinctly pronounced:
- For slits, there are secondary maxima between two primary maxima.
- With an increase in the number of slits to :
- the primary maxima will become thinner and sharper
- The secondary maxima will become unimporant
- The intensity of the central maximum is proportional to .
- used in spectroscopy to measure the wavelength of light.
- Has rulings, which are slits, which help determine the slit seperation.
- lines/rulings per millimetre corresponds to a slit seperation.
- Since we know that the condition for constructive interference is , we can use this to calculate the wavelength of light.
- Upon reflection on a surface with a refractive index greater than the medium a ray is already in, the ray undergoes a phase change of .
- If is the thickness of the film, the condition for constructive interference, when only one phase change occurs (light is reflected off a medium with a greater refractive index only once):
- : thickness of the film.
- : refractive index.
- : an integer.
Note: It is similar to the condition for destructive interference (), but it is the condition for constructive interference in this case because of the phase shift of , resulting in crests becoming troughs and troughs becoming crests.
- The condition for destructive interference, when only one phase change occurs(light is reflected off a medium with a greater refractive index only once):
- The condition for constructive interference when there is no or two phase changes (light is reflected off a medium with a greater refractive index twice):
- The condition for destructive interference when there is no or two phase changes (light is reflected off a medium with a greater refractive index twice):
- The angular separation of two objects is given by where is the distance between the objects, and is the distance between the observer and the objects.
- According to the Rayleigh criterion, resolution is possible when the angular separation is greater than the angle fo the first diffraction minimum:
- For a circular slit, the following criteria is used:
In a diffraction grating
- : resolving power of the grating
- : The average of the two wavelengths to be resolved
- : difference in the wavelengths that are to be resolved
- The doppler effect is the change in the observed frequency of a wave which happens whenever there is relative motion between the source and the observer.
- If the wavelength of the light decreases, the source is moving towards the observer, which is called a blueshift.
- If the wavelength of the light increases, the source is moving away from the observer, which is called a redshift.
- If a source is moving towards an observer with velocity :
- If an observer is moving towards a source with velocity :
- If the speed of the observer is small compared to the speed of light, then:
- K. A. Tsokos - Physics for the IB Diploma, Sixth Edition
- Class Notes