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IB PHYSICS Topic 4: Oscillations and Waves

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

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IB PHYSICS Topic 4: Oscillations and Waves

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

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