4.1 Oscillations

• Periodic Motion:

  • Motion that repeats itself after equal intervals of time. 

    • Examples include the motion of a loaded spring, an object moving in a circle, and a simple pendulum.

• Terms Related to Periodic Motion:

  • Amplitude (A): Maximum displacement from the equilibrium position.

  • Time period (T): Time taken for a complete oscillation.

  • Frequency (f): Number of oscillations per unit time. (f = 1/T)

  • Angular frequency (𝓌): Equivalent of frequency. (𝓌 = 2(π)f)

  • Phase (θ): If the motion starts away from equilibrium, it leads or lags by θ.

  • Isochronous oscillations: Maintain a constant time period regardless of amplitude changes.

• Simple Harmonic Motion (SHM):

  • A type of periodic motion where the restoring force is proportional to the negative displacement from the equilibrium position. 

    • Examples include a spring-loaded with a mass and a simple pendulum with a small amplitude.

  • Equation defining SHM: a - kx where a is accelerated, k is a constant, and x is displacement.

    • Units of constant k: 

      • K = - a/x, so the unit of k is m x s^-2 / m = s^-2.

    • Difference in oscillations of two systems S1 and S2.

      • If S1 has frequency f, S2 with 4k frequency has a frequency of √4f = 2f

• Describing Simple Harmonic Motion:

  • Equation for SHM: x  = Asin(2πft + θ) = Asin(wt + θ), where θ is π/2

• Velocity (v):

  • v = (dx)/(dt) = 𝓌Acos(𝓌t + θ)

• Acceleration(a):

  • a = (dv)/(dt) = -𝓌^2(A)(sin(𝓌t+θ))

• Phase difference between displacement-time graphs:

  • About 25 seconds

  • 0.79 radians

• Circular Motion and SHM: 

  • The projection of an object in circular motion on a diameter follows simple harmonic motion.

• Energy Changes in Simple Harmonic Motion:

  • Kinetic energy: KE= ½(m𝓌^2A^2)

    • Total energy remains constant in the absence of dissipative forces.

• Waves and their types:

  • Mechanical waves: require a material medium to travel

  • Electromagnetic waves: can travel through a vacuum.

• Describing Waves:

  • Wavefront: A surface perpendicular to the direction of wave travel.

  • Amplitude (A): Maximum displacement from equilibrium.

  • Wavelength (⅄): Shortest distance between two points in phase on a wave.

  • Period (T): Time for a complete wavelength to pass a fixed point.

  • Frequency (f): Number of wavelengths passing through a fixed point per unit time.

    • f = 1/T

4.2 Traveling waves

• Transverse waves: Direction of vibration perpendicular to the direction of propagation.

• Longitudinal waves: Direction of vibration parallel to the direction of propagation. 

• Wave Equation:

  • The velocity of a wave (c) is given by c = f⅄

• Electromagnetic waves:

  • Travel with varying electric and magnetic fields at 3 x 10^8 m/s in a vacuum.

4.3 Wave characteristics

• Intensity of Waves:

  • Intensity (I) is power received per unit area. I = (P)/(4πr^2) and is proportional to the square of amplitude (A^2).

  • Example:

    • Intensity at 120m from source: 3 x 10^-6 W/m^2.

• Principle of Superposition:

  • When two waves meet, the total displacement is the vector sum of their individual displacements.

• Polarization:

  • Restriction of oscillation direction to a plane perpendicular to the direction of propagation. Result: Plane-polarized light.

• Malus’s Law:

  • Intensity (I) transmitted by an analyzer is proportional to cos^2(θ) where θ is the angle between the polarizer and the analyzer.

4.4 Wave Behaviour

• Laws Of Reflection And Refraction:

  • Incident, reflected, and refracted rays, and normal lie on the same plane.

  • The angle of incidence equals the angle of reflection.

  • (sinθ1)/(sinθ2) = 1/n (Snell’s Law)

• Reversibility of Light:

  • (sinθ1)/(sinθ2) = 1/n_2 for light going from medium 1 to medium 2, and (sinθ1)/(sinθ2) = 1/n_1 for light traveling in the opposite direction.

• Critical Angle And Total Internal Reflection:

  • The angle of incidence for which the angle of refraction reaches the right angle is the critical angle. 

  • Total internal reflection occurs when the angle of incidence is greater than the critical angle.

• Double-Slit Interference:

  • Two coherent sources create interference patterns. Constructive interference occurs at nλ and is destructive at (n + ½)λ.

  • Example:

    • The path difference at point P is 7λ. The nature of the fringe at P is bright, and there are 7 dark fringes between O and P.

• Diffraction:

  • Wave passed through a narrow gap forms bright and dark fringes. Angular position of minima given by θ = (nλ/a).

  • Example:

    • Path difference at point P is 7λ. 

    • The nature of fringe at P is bright, and there are 7 dark fringes between O and P.

• Interference With Multiple Slits:

  • More slits result in sharper and more intense maxima and minima.

• Dispersion:

  • Different wavelengths of light refract at different angles. White light disperses into its constituent wavelengths.

• Resolution:

  • Rayleigh's criterion states two points are just resolved if the central maximum of the first point falls on the first minimum of the second point.

• Diffraction Grating:

  • For a grating with N slits, R = λ/change in λ = mN

• Reflection Of Light Off Thin Films:

  • Reflected light undergoes a phase change of 180∘ if reflected off a denser medium. 

  • A thin film of thickness t, refractive index n, and incident wavelength λ exhibits interference.

• Doppler Effect In Light:

  • The change in frequency of the light wave is (v/c)(f_0).

• Water Waves:

  • Follow laws similar to light. Exhibit reflection, refraction, interference, and diffraction.

• Wave Propagation:

  • Wavefront consists of infinite new disturbance centers. 

  • Successive wavefronts result from wavelets from these disturbances.

• Reflection Of Water Wave:

  • When a wave hits a barrier, it behaves as if a similar wave is coming from the barrier in the opposite direction.

• Doppler Effect In Sound:

  • The frequency of a moving source changes for an observer at rest or moving toward/away from the source.

4.5 Standing waves

• Boundary Conditions:

  • Reflected off a fixed boundary suffers a phase change of 180∘. 

  • No change in the phase of a free boundary.

• Standing Waves:

  • Formed when two waves of equal amplitude and frequency traveling in opposite directions are superimposed. 

  • Positions of crests and troughs do not change with time.

• Nodes and Antinodes:

  • Nodes are points with zero displacement, antinodes are points with maximum displacement.

• Harmonics On A String:

  • The string is tied at one end and connected to a vibration generator at the other. 

    • Harmonics formed with increasing loops at n times the frequency of the first harmonic.

• Displacement of string at different times:

  • Quarter of a cycle: t = 1/4f

  • Half of a cycle: t = 1/2f

• Frequency of vibration of the spring:

  • Wavelength 2L, wave velocity 240 m/s, frequency 120 Hz.

• Harmonics In A Pipe:

  • Harmonics formed with one end open or both ends open. Nodes form at closed ends and antinodes at open ends.

• Explanation regarding refraction of light: 

  • The speed of light is faster in a vacuum than in water, bending away from normal.

• The critical angle for total internal reflection:

  • sinፀ_c = 1/n

    • where n is the relative refractive index of denser material with respect to rarer material.

• Frequency of the first harmonic if both ends are open: 

  • Twice the frequency of the first harmonic when one end is closed