SHM equations

General equation:

For a system starting at equilibrium:

For a system starting at maximum displacement:

Angular frequency:

Equation for simple harmonic oscillators - Khan Academy

Energy in SHM systems

In SHM there is energy interchange between PE and KE, however the total energy remains constant.

Variation of energy with displacement:

Variation of energy with time:

Energy In a Simple Harmonic Oscillator - video explanation

Graph of intensity against angle

The single-slit equation

Single-slit with monochromatic and white light

The angular width of the central maximum and the angular separation of successive secondary maxima depend on the wavelenght of the light. This is why the edges of the diffraction pattern are colored.

Video explanation

Intensity variation with the double-slit

Double-slit diffraction pattern

The double-slit interference pattern is a superposition of the relative intensity without diffraction for a double-slit and the variation of intensity relative to angle of a single-slit.

Young's double slit introduction - Khan Academy
Double & Single Slit Experiments and Diffraction Gratings
Hardy's Paradox (optional)

Multiple-slit interference

The effect of modulation increases with the number of slits. This causes the fringes to be narrower and their intensity being proportional to the square of the number of slits.

Video demonstration
Video explanation

Diffraction grating

Diffraction gratings are the consequence of the effect on the interference pattern when the number of slits is increased. They produce optical spectra and contain a large number of parallel lines (slits).

Diffraction grating video explanation

Grating spacing and number of lines per mm

N must be converted to the number of lines per meter (multiply by 1000).

Interference by division of amplitude

In an amplitude-division system, a beam splitter is used to divide the light into two beams travelling in different directions, which are then superimposed to produce the interference pattern. The Michelson interferometer and the Mach–Zehnder interferometer are examples of amplitude-division systems.

Thin film interference

Thin-film interference is a natural phenomenon in which light waves reflected by the upper and lower boundaries of a thin film interfere with one another, either enhancing or reducing the reflected light.

Thin film interference is the wave phenomenon that is responsible for the formation of (for example) regions of different color when white light is reflected from a thin film of oil floating on water.

Example with a soap bubble:

Waves reflected by the film

Video explanation

Thin-film interference caused by ITO defrosting coating on an Airbus cockpit window:

Diffraction and resolution

Resolution is the ability of an imaging system to be able to produce two separate distinguishable images of two separate objects.

Rayleigh criterion states that two sources are resolved if the principal maximum from one diffraction pattern is no closer than the first minimum of the other pattern.

The limit to resolution is when the principal maximum of the diffraction pattern from one source lies on the first minimum diffraction pattern from the second source (and vice versa).

Resolution equation

Examples:

Resolution IB Physics - video explanation

Resolvance of diffraction gratings

The doppler effect with sound waves

The Doppler effect is observed whenever the source of waves is moving with respect to an observer. The Doppler effect can be described as the effect produced by a moving source of waves in which there is an apparent upward shift in frequency for observers towards whom the source is approaching and an apparent downward shift in frequency for observers from whom the source is receding.

Source moving towards observer at rest:

Source moving away from observer at rest:

Observer moving towards stationary source:

Observer moving away from stationary source:

Where:

Video explanation

The Doppler effect with light

Because electromagnetic waves do not need a medium, the Doppler effect for EM waves is simply a relative-velocity phenomenon:

^*This equation should only be used when the velocity of the observer is much smaller than the speed of light

Red Shift and Doppler Effect

1. In order that the interference between the waves emitted by two light sources can be observed, it is essential that the sources must emit waves that

A. have the same amplitude

B. are in phase

C. have the same color

D. have a constant phase difference between them

2. A 0.45 kg object causes a vertical spring to stretch 0.16 m. What is the period of this combination when it is oscillating?

A. 0.72 s

B. 0.80 s

C. 0.35 s

D. 1.9 s

3. Which one of the following diagrams best represents wavefronts produced by a source of sound of constant frequency as it moves at constant speed towards a stationary observer at O?

A.

B.

C.

D.

4. Two binary stars emit radio waves of wavelength 6.0 ×10⁻² m. The waves are received by a radio telescope whose collecting dish has a diameter of 120 m. The two stars are just resolved if their minimum angular separation in radians is of the order of

A. 2 ×10⁴

B. 2 ×10^>2

C. 5 ×10^–2

D. 5 ×10^>–4

5. Light from a double slit arrangement produces bright and dark fringes on a screen in the region near point P, as indicated below. The light from the two slits has equal amplitudes on reaching point P. Which one of the following gives the change, if any, in the appearance of the bright and the dark fringes when the amplitude of the light wave from one slit is reduced?

A.

B.

C.

D.

6. A sound emitting source moves along a straight line with speed v relative to an observer at rest. The speed of sound relative to the medium is c. The observer measures the speed of sound emitted by the source as

A. c

B. c+v

C. c-v

D. v-c

7. A source of sound emits waves of wavelength λ, period T and speed v when at rest. The source moves away from a stationary observer at speed V, relative to the observer. The wavelength of the sound waves, as measured by the observer is

A. λ + vT

B. λ – vT

C. λ + VT

D. λ – VT

8. A particle performs simple harmonic oscillations with amplitude 0.1 mm and frequency 100 Hz. What is the maximum acceleration of this particle (in m s⁻²)

A. 0.2π

B. 0.4π

C. 2π²

D. 4π²

9. Light is incident on N very thin parallel slits and an interference pattern is formed on a screen a distance away. The number of slits is increased while the separation of two consecutive slits stays the same. Which is correct as N increases?

A. the number of secondary maxima decreases

B. the intensity of the secondary maxima increases

C. the primary maxima become narrower

D. the distance between the central maximum and the first primary maximum to the side increases

10. Which of the following graphs shows the variation with time of the potential energy of a particle undergoing simple harmonic oscillations with a period of 1.0 s?

A.

B.

C.

D.

Number of correct answers: