Comprehensive Proportionality Patterns in Oscillations and Waves
| Relationship/Concept | Proportionality Pattern | Governing Formula/Concept | Explanation |
| Fundamental Concepts of Oscillation | |||
| Frequency vs. Period | Frequency is inversely proportional to the period. (v ∝ 1/T) | v = 1/T | If the time for one oscillation (period) is doubled, the number of oscillations per second (frequency) is halved. |
| Angular Frequency vs. Period | Angular frequency is inversely proportional to the period. (ω ∝ 1/T) | ω = 2π / T | A system with a long period of oscillation has a small angular frequency. |
| Simple Harmonic Motion (SHM): Dynamics and Kinematics | |||
| Restoring Force vs. Displacement | Restoring force is directly proportional to displacement (and in the opposite direction). (F ∝ -x) | F = -kx | The farther an object is displaced from its equilibrium, the stronger the force pulling it back. |
| Acceleration vs. Displacement | Acceleration is directly proportional to displacement (and in the opposite direction). (a ∝ -x) | a(t) = -ω²x(t) | The magnitude of acceleration is zero at the equilibrium position and maximum at the extreme points of displacement. |
| Maximum Velocity vs. Amplitude | Maximum velocity is directly proportional to the amplitude. (v_max ∝ A) | v_max = ωA | If the amplitude of oscillation doubles, the maximum speed reached by the object also doubles (for a constant ω). |
| Maximum Acceleration vs. Amplitude | Maximum acceleration is directly proportional to the amplitude. (a_max ∝ A) | a_max = ω²A | Doubling the amplitude of oscillation doubles the maximum acceleration experienced by the object (for a constant ω). |
| Energy in Simple Harmonic Motion | |||
| Total Energy vs. Amplitude | Total energy is proportional to the square of the amplitude. (E ∝ A²) | E = (1/2)kA² | If the amplitude of oscillation is doubled, the total energy of the system increases by a factor of four. |
| Kinetic Energy vs. Velocity | Kinetic energy is proportional to the square of the velocity. (K ∝ v²) | K = (1/2)mv² | Kinetic energy is maximum when velocity is maximum (at the equilibrium position) and zero at the extreme points where velocity is momentarily zero. |
| Potential Energy vs. Displacement | Potential energy is proportional to the square of the displacement. (U ∝ x²) | U = (1/2)kx² | Potential energy is maximum at the extreme points of displacement and zero at the equilibrium position. |
| Specific Oscillating Systems | |||
| Period of Spring-Mass System vs. Mass | The period is proportional to the square root of the mass. (T ∝ √m) | T = 2π√(m/k) | To double the period of oscillation for a given spring, you must increase the mass by a factor of four. |
| Period of Spring-Mass System vs. Spring Constant | The period is inversely proportional to the square root of the spring constant. (T ∝ 1/√k) | T = 2π√(m/k) | A stiffer spring (larger k) will cause the mass to oscillate with a shorter period. |
| Period of Simple Pendulum vs. Length | The period is proportional to the square root of its length. (T ∝ √L) | T = 2π√(L/g) | To double the time it takes for a pendulum to complete one swing, its length must be quadrupled. The period is independent of the bob’s mass. |
| Period of Simple Pendulum vs. Gravity | The period is inversely proportional to the square root of the acceleration due to gravity. (T ∝ 1/√g) | T = 2π√(L/g) | A pendulum of a given length will have a longer period (swing slower) on the Moon than on Earth because the Moon’s gravity is weaker. |
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| Relationship/Concept | Proportionality Pattern | Governing Formula/Concept (Chain of Logic) | Explanation/Context |
| Core SHM & Energy Relationships | |||
| Period of Spring-Mass System vs. Spring Stiffness | Period is inversely proportional to the square root of the spring constant. (T ∝ 1/√k) | T = 2π√(m/k) | A stiffer spring (higher k) results in a faster oscillation and therefore a shorter period. |
| Stored Potential Energy vs. Spring Stiffness | For a given displacement, potential energy is directly proportional to the spring constant. (U ∝ k) | U = (1/2)kx² | A stiffer spring stores more energy for the same amount of stretch or compression. |
| Kinetic Energy vs. Potential Energy | Kinetic energy is inversely related to potential energy during oscillation. | E_total = K + U = constant | As the oscillator moves away from equilibrium, kinetic energy is converted into potential energy. As it moves toward equilibrium, the reverse happens. |
| Frequency of a Pendulum vs. Length | Frequency is inversely proportional to the square root of the length. (f ∝ 1/√L) | f = 1/T and T = 2π√(L/g) | A short pendulum swings back and forth more frequently (has a higher pitch if it makes a sound) than a long pendulum. |
| Wave Properties (Derived from Oscillations) | |||
| Frequency vs. Wavelength (for a constant wave speed) | Frequency is inversely proportional to wavelength. (f ∝ 1/λ) | v_wave = fλ, so f = v_wave/λ | For sound waves traveling in air (constant speed), high-frequency sounds (high pitch) have short wavelengths. |
| Wave Speed in a String vs. Tension | Wave speed is proportional to the square root of the tension. (v ∝ √F_T) | v = √(F_T / μ) (where μ is mass per unit length) | Tightening a guitar string (increasing tension) increases the speed of the wave, which in turn increases its fundamental frequency (pitch). |
| Fundamental Frequency of a String vs. Tension | Frequency is proportional to the square root of the tension. (f ∝ √F_T) | Combines f ∝ v (from v=fλ) and v ∝ √F_T | This is why tuning a guitar by tightening the strings raises the pitch of the notes. |
| Quantum & Electromagnetic Wave Relationships | |||
| Energy of a Photon vs. Frequency | Energy is directly proportional to frequency. (E ∝ f) | E = hf (Planck’s relation) | This is a cornerstone of quantum mechanics. Higher frequency electromagnetic waves, like UV light, carry more energy per photon than lower frequency waves, like radio waves. |
| Energy of a Photon vs. Wavelength | Energy is inversely proportional to wavelength. (E ∝ 1/λ) | E = hf and f = c/λ → E = hc/λ | This is a highly useful derived relationship. High-energy gamma rays have extremely short wavelengths, while low-energy radio waves have very long wavelengths. |
| Momentum of a Photon vs. Frequency | Momentum is directly proportional to frequency. (p ∝ f) | p = E/c and E = hf → p = hf/c | Photons of higher frequency light will impart more momentum upon impact. |
| de Broglie Wavelength of a Particle vs. Momentum | Wavelength is inversely proportional to momentum. (λ ∝ 1/p) | λ = h/p | This is the principle of wave-particle duality. A fast-moving particle (high momentum) exhibits a shorter wavelength. This is fundamental to electron microscopes. |
| Damped & Driven Oscillations | |||
| Amplitude of Damped Oscillation vs. Time | Amplitude decays exponentially with time. | A(t) = A₀e^(-bt/2m) | The energy of the oscillator is lost to friction or drag, causing the amplitude of swings to get progressively smaller. |
| Rate of Energy Loss vs. Damping Factor | The rate of energy loss is proportional to the square of the velocity and directly proportional to the damping constant. (P_loss ∝ b v²) | F_damping = -bv, Power = F · v | A thicker fluid (higher b) or faster motion will dissipate the oscillator’s energy more quickly. |
| Amplitude at Resonance vs. Damping Factor | The maximum amplitude at resonance is inversely proportional to the damping factor. (A_max ∝ 1/b) | Derived from the amplitude equation for driven oscillators. | A system with very low damping (like a crystal glass) can shatter from resonance because the amplitude can grow to destructive levels. |
| Sharpness of Resonance Peak vs. Damping | The sharpness of the resonance peak is inversely proportional to damping. | A high “Q factor” (Quality factor) means low damping and a sharp, narrow resonance peak. | Systems with low damping are very sensitive to being driven at a specific frequency (e.g., a radio tuner). |
Proportionality Comparisons for MCQs in Oscillations
This table shows the ratio between two systems (System 2 vs. System 1) when a single parameter is changed.
| System / Property | Parameter Changed | Resulting Proportional Change | Ratio Formula for Calculation |
| 1. Spring-Mass System | |||
| Period (T) | Mass (m) is changed. | Period is proportional to the square root of mass. | T₂ / T₁ = √(m₂ / m₁) |
| Period (T) | Spring Constant (k) is changed. | Period is inversely proportional to the square root of the spring constant. | T₂ / T₁ = √(k₁ / k₂) |
| Frequency (v) | Mass (m) is changed. | Frequency is inversely proportional to the square root of mass. | v₂ / v₁ = √(m₁ / m₂) |
| Frequency (v) | Spring Constant (k) is changed. | Frequency is proportional to the square root of the spring constant. | v₂ / v₁ = √(k₂ / k₁) |
| 2. Simple Pendulum | |||
| Period (T) | Length (L) is changed. | Period is proportional to the square root of length. | T₂ / T₁ = √(L₂ / L₁) |
| Period (T) | Gravity (g) is changed (e.g., Earth vs. Moon). | Period is inversely proportional to the square root of gravity. | T₂ / T₁ = √(g₁ / g₂) |
| Frequency (v) | Length (L) is changed. | Frequency is inversely proportional to the square root of length. | v₂ / v₁ = √(L₁ / L₂) |
| 3. Energy in SHM | |||
| Total Energy (E) | Amplitude (A) is changed. | Energy is proportional to the square of the amplitude. | E₂ / E₁ = (A₂ / A₁)² |
| Total Energy (E) | Frequency (v) or Angular Frequency (ω) is changed. | Energy is proportional to the square of the frequency. | E₂ / E₁ = (v₂ / v₁)² = (ω₂ / ω₁)² |
| Total Energy (E) | Spring Constant (k) is changed (for the same amplitude). | Energy is directly proportional to the spring constant. | E₂ / E₁ = k₂ / k₁ |
| Amplitude (A) | Energy (E) is changed (for the same oscillator). | Amplitude is proportional to the square root of the energy. | A₂ / A₁ = √(E₂ / E₁) |
| 4. Kinematics of SHM (Velocity & Acceleration) | |||
| Max Velocity (v_max) | Amplitude (A) is changed (at same frequency). | Max velocity is directly proportional to the amplitude. | v_max₂ / v_max₁ = A₂ / A₁ |
| Max Velocity (v_max) | Frequency (v) is changed (at same amplitude). | Max velocity is directly proportional to the frequency. | v_max₂ / v_max₁ = v₂ / v₁ |
| Max Acceleration (a_max) | Amplitude (A) is changed (at same frequency). | Max acceleration is directly proportional to the amplitude. | a_max₂ / a_max₁ = A₂ / A₁ |
| Max Acceleration (a_max) | Frequency (v) is changed (at same amplitude). | Max acceleration is proportional to the square of the frequency. | a_max₂ / a_max₁ = (v₂ / v₁)² |
| Max Acceleration (a_max) | Displacement (x) from equilibrium is changed. | Acceleration is directly proportional to the displacement. | a₂ / a₁ = x₂ / x₁ |
Here are the key variations of the question you provided, structured in a format perfect for MCQ practice.
Variations of SHM Formulas for MCQs
The table is organized by the quantity being asked for. Each row represents a potential multiple-choice question.
| If you are given… | And you are asked for… | The answer is… | How to derive it |
| Category 1: Finding Maximum Speed (v_max) | |||
| Amplitude (A) and Period (T) | Maximum Speed (v_max) | 2πA / T | Start with v_max = ωA and substitute ω = 2π/T. |
| Amplitude (A) and Frequency (v) | Maximum Speed (v_max) | 2πvA | Start with v_max = ωA and substitute ω = 2πv. |
| Amplitude (A), Mass (m), and Spring Constant (k) | Maximum Speed (v_max) | A√(k/m) | Start with v_max = ωA and substitute ω = √(k/m). |
| Total Energy (E) and Mass (m) | Maximum Speed (v_max) | √(2E/m) | Use the energy formula E = ½mv_max² and solve for v_max. |
| Max Acceleration (a_max) and Amplitude (A) | Maximum Speed (v_max) | √(a_max ⋅ A) | Combine v_max² = ω²A² and a_max = ω²A to get v_max² = a_max ⋅ A. |
| Max Acceleration (a_max) and Angular Frequency (ω) | Maximum Speed (v_max) | a_max / ω | Rearrange a_max = ω ⋅ v_max (since ω²A = ω ⋅ ωA). |
| Category 2: Finding Maximum Acceleration (a_max) | |||
| Amplitude (A) and Period (T) | Maximum Acceleration (a_max) | 4π²A / T² | Start with a_max = ω²A and substitute ω = 2π/T. |
| Amplitude (A) and Frequency (v) | Maximum Acceleration (a_max) | 4π²v²A | Start with a_max = ω²A and substitute ω = 2πv. |
| Max Speed (v_max) and Period (T) | Maximum Acceleration (a_max) | 2πv_max / T | Combine a_max = ωv_max and ω = 2π/T. |
| Max Force (F_max) and Mass (m) | Maximum Acceleration (a_max) | F_max / m | Use Newton’s second law, F = ma. |
| Spring Constant (k), Amplitude (A), and Mass (m) | Maximum Acceleration (a_max) | kA / m | Combine F_max = kA and F_max = ma_max, then solve for a_max. |
| Category 3: Finding Period (T) or Frequency (v) | |||
| Mass (m) and Spring Constant (k) | Period (T) | 2π√(m/k) | Start with ω = √(k/m) and use T = 2π/ω. |
| Max Speed (v_max) and Amplitude (A) | Period (T) | 2πA / v_max | Rearrange the formula v_max = 2πA / T. |
| Max Acceleration (a_max) and Amplitude (A) | Period (T) | 2π√(A/a_max) | Start with a_max = ω²A, solve for ω, then find T. |
| Max Acceleration (a_max) and Max Speed (v_max) | Period (T) | 2πv_max / a_max | Start with a_max = ωv_max, solve for ω = a_max/v_max, then find T. |
| Category 4: Finding Total Energy (E) | |||
| Mass (m), Amplitude (A), and Period (T) | Total Energy (E) | 2mπ²A² / T² | Start with E = ½mω²A² and substitute ω = 2π/T. |
| Spring Constant (k) and Amplitude (A) | Total Energy (E) | ½kA² | This is a fundamental energy formula for a spring. |
| Mass (m) and Max Speed (v_max) | Total Energy (E) | ½mv_max² | The total energy equals the maximum kinetic energy. |