Oscillations & Waves (Unit 03) - Study Guide

Core Definitions:

• • Simple Harmonic Motion (SHM): Periodic motion where the acceleration is directly proportional to the displacement from the equilibrium position and is always directed towards the equilibrium.
• • Damped Oscillations: Oscillations where the amplitude decreases over time due to resistive forces (air resistance, friction).
• • Resonance: The phenomenon where a system oscillates with maximum amplitude when the frequency of an external periodic force equals the natural frequency of the system.

Equations:

• • SHM Acceleration: a = -ω² x (a: acceleration, ω: angular frequency, x: displacement. Negative sign shows direction is opposite to displacement)
• • SHM Velocity: v = ±ω√(A² - x²) (A: amplitude)
• • Simple Pendulum Time Period: T = 2π√(l/g)
• • Spring-mass system Time Period: T = 2π√(m/k)

Key Points:

• • At the equilibrium position (x = 0), the velocity of the object is maximum and the acceleration is zero.
• • At the extreme ends (x = A), the velocity is zero and the acceleration is maximum.

Core Definitions:

• • Transverse Waves: Waves in which the particles of the medium vibrate perpendicular to the direction of wave propagation (e.g. water ripples, light).
• • Longitudinal Waves: Waves in which the particles of the medium vibrate parallel to the direction of wave propagation (e.g. sound waves).
• • Wavelength (λ): The distance between two consecutive particles in the same phase.

Equations:

Wave Equation: v = fλ (v: wave speed, f: frequency, λ: wavelength)

Key Points:

• • Waves display 4 phenomena: Reflection, Refraction, Diffraction, and Interference.
• • Principle of Superposition: The resultant displacement is the vector sum of individual wave displacements.

Core Definitions:

• • Nodes: Points in a stationary wave that remain permanently at rest (zero displacement).
• • Antinodes: Points in a stationary wave where displacement is maximum.
• • Fundamental Note: The lowest frequency note that a system can produce (f₀).

Equations:

• • Speed of transverse wave on string: v = √(T/m) (T: tension, m: mass per unit length)
• • Fundamental frequency of string: f₀ = 1/(2l) * √(T/m)
• • Closed Pipes: f_n = nv / 4l (n = 1, 3, 5...)
• • Open Pipes: f_n = nv / 2l (n = 1, 2, 3...)

Key Points:

• • Closed pipes produce only odd harmonics (1, 3, 5...).
• • Open pipes produce all harmonics (1, 2, 3, 4...).

Core Definitions:

• • Doppler Effect: The apparent change in frequency of sound heard due to the relative motion between the sound source and the observer.
• • Loudness: The subjective sensation of sound depending on the physical intensity of the wave.
• • Pitch: The subjective sensation of sound depending on the physical frequency of the wave.

Equations:

• • Doppler Formula: f = ((v ± u₀) / (v ∓ u_s)) * f₀ (v: speed of sound, u₀: speed of observer, u_s: speed of source)
• • Sound Intensity Level: β = 10 log₁₀(I / I₀) dB

Key Points:

• • Frequency increases as observer and source move closer, and decreases as they move apart.
• • The human audio frequency range is 20 Hz to 20,000 Hz.

Core Definitions:

• • Critical Angle (c): The angle of incidence in the denser medium for which the angle of refraction in the rarer medium is 90°.
• • Total Internal Reflection (TIR): The complete reflection of light back into the denser medium when the angle of incidence exceeds the critical angle.

Equations:

• • Snell's Law: n₁ sin i₁ = n₂ sin i₂
• • Critical Angle: sin c = 1 / n
• • Lens Formula: 1/v - 1/u = 1/f

Key Points:

• • Convex lenses act as converging lenses, while concave lenses act as diverging lenses.
• • Minimum deviation in a prism occurs when the angle of incidence equals the angle of emergence.

Core Definitions:

• • Myopia (Short-sightedness): Difficulty seeing distant objects clearly because the image forms in front of the retina.
• • Hypermetropia (Long-sightedness): Difficulty seeing near objects clearly because the image forms behind the retina.
• • Angular Magnification (m): The ratio of the angle subtended by the image to that subtended by the object at the unaided eye.

Equations:

• • Simple Microscope Magnification: m = D / u
• • Telescope Magnification (Normal adjustment): m = f_o / f_e

Key Points:

• • Myopia is corrected using a concave lens; Hypermetropia is corrected using a convex lens.