Oscillations and Waves

Definition

Periodic Motion is any motion that repeats itself at regular intervals of time (e.g., the Earth revolving around the sun). Oscillatory Motion is a specific type of periodic motion where an object moves back and forth about a central, stable position called the equilibrium position.

Key Concepts

• Equilibrium Position: The point where the net force acting on the object is zero.
• Amplitude (A): The maximum distance the object moves away from the equilibrium position.
• Time Period (T): The time taken to complete one full oscillation.
• Frequency (f): The number of oscillations completed in one second (f = 1/T).

Examples

• A child on a swing moving back and forth.
• The pendulum of a wall clock.

Meaning and Definition

SHM is a special type of oscillation where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction.

Mathematical Formula: F = -kx (where k is the spring constant).

Velocity and Acceleration in SHM

• Displacement (x): x = A sin(ωt)
• Velocity (v): v = ω√(A² - x²) (Max at equilibrium, zero at extremes).
• Acceleration (a): a = -ω²x (Max at extremes, zero at equilibrium).

Energy in SHM

• Kinetic Energy (KE): ½ m ω² (A² - x²)
• Potential Energy (PE): ½ m ω² x²
• Total Energy: KE + PE = ½ m ω² A² (Remains constant in ideal SHM).

Common Systems

• Spring-Mass System: T = 2π√(m/k)
• Simple Pendulum: T = 2π√(l/g)

• Free Oscillations: Vibration at natural frequency without external interference.
• Damped Oscillations: Amplitude gradually decreases due to resistive forces (friction/air drag).
• Forced Oscillations: When an external periodic force is applied to maintain motion.

Resonance

Resonance occurs when the frequency of an external driving force matches the natural frequency of the system, causing a massive increase in amplitude.

Transverse vs. Longitudinal Waves

Key Formulas & Concepts

• Wave Speed: v = fλ (Frequency × Wavelength)
• Superposition: Resultant displacement is the sum of individual displacements.
• Beats: Periodic rise/fall in volume when two slightly different frequencies interfere.
• Doppler Effect: Change in frequency due to relative motion between source and observer.

Question: A mass of 0.5 kg is attached to a spring with a spring constant k = 200 N/m. The mass is pulled 0.1 m from equilibrium and released. Find the total energy and the time period.

Step 1: Identify Givens
- m = 0.5 kg
- k = 200 N/m
- A = 0.1 m

Step 2: Calculate Total Energy (E)
E = 1/2 * k * A^2
E = 1/2 * 200 * (0.1)^2
E = 100 * 0.01 = 1 Joule

Step 3: Calculate Time Period (T)
T = 2π * sqrt(m/k)
T = 2π * sqrt(0.5/200)
T = 2π * sqrt(0.0025)
T = 2π * 0.05
T ≈ 0.314 seconds

Final Answer: Energy = 1 J, T ≈ 0.314 s
        

Engineers use SHM and Damping to create comfortable cars. A spring undergoes SHM when hitting a bump, but without "shock absorbers" (damping devices), the car would bounce forever (Free Oscillation). Damping converts kinetic energy into heat, stopping the bounce. Engineers also tune the system to avoid Resonance, ensuring road vibrations don't match the car's natural frequency and cause structural failure.

1. In SHM, where is the velocity of the particle maximum?
   a) At the extremes | b) At the equilibrium position | c) Midway | d) Constant
   Answer: b
2. The relationship between frequency (f) and time period (T) is:
   a) f = T | b) f = 1/T | c) f = 1/T² | d) f = 2πT
   Answer: b
3. Which of the following is a longitudinal wave?
   a) Light | b) Water | c) Sound | d) String
   Answer: c
4. The restoring force in SHM is proportional to:
   a) Velocity | b) Acceleration | c) Displacement | d) Time
   Answer: c
5. When driving frequency equals natural frequency, we observe:
   a) Damping | b) Resonance | c) Interference | d) Reflection
   Answer: b
6. The unit of frequency is:
   a) Second | b) Meter | c) Hertz | d) Newton
   Answer: c
7. In a transverse wave, the particles move:
   a) Parallel | b) Perpendicular | c) In circles | d) No movement
   Answer: b
8. The Doppler Effect is associated with:
   a) Light and Sound | b) Only Sound | c) Only Light | d) Magnetism
   Answer: a
9. If the length of a pendulum is increased, its time period will:
   a) Increase | b) Decrease | c) Stay same | d) Zero
   Answer: a
10. Damping in an oscillator causes the amplitude to:
    a) Increase | b) Stay constant | c) Decrease | d) Flip
    Answer: c

1. Define Simple Harmonic Motion and give its mathematical condition.
   Answer: SHM is a periodic motion where the restoring force is proportional to displacement and acts towards equilibrium (F = -kx).
2. Distinguish between Transverse and Longitudinal waves.
   Answer: Transverse waves move perpendicular to wave direction (e.g., Light); Longitudinal waves move parallel (e.g., Sound).
3. What is the difference between Free and Forced oscillations?
   Answer: Free oscillations happen at natural frequency without external force; Forced oscillations are driven by an external periodic force.
4. Explain Resonance with a real-life example.
   Answer: Resonance is when external force matches natural frequency, causing large amplitude. Example: A singer breaking glass.
5. Derive the expression for the time period of a simple pendulum.
   Answer: Using F = -mg sinθ ≈ -mgθ and equating to SHM force, we get T = 2π√(l/g).
6. How does the frequency of sound change as a source approaches an observer?
   Answer: The frequency increases due to the Doppler Effect.
7. Why doesn't a pendulum's period depend on amplitude (for small angles)?
   Answer: Because for small angles, sinθ ≈ θ, keeping the restoring force linear.
8. Define 'Beats'.
   Answer: Periodic variations in intensity when two waves of slightly different frequencies interfere.
9. Calculate speed if f = 50 Hz and λ = 2m.
   Answer: v = fλ = 50 * 2 = 100 m/s.
10. State the Principle of Superposition.
    Answer: When waves overlap, the resultant displacement is the algebraic sum of individual displacements.

• Transverse = Top-to-bottom (Perpendicular).
• Longitudinal = Line (Parallel).
• SHM Energy: "At the center, KE is King; At the ends, PE is Prime."
• Wave Speed: v = fλ.

Modern Industry: Ultrasound (Medical), Radio Waves (Telecom), Seismology (Earthquakes).