The Rhythm of Physics: Mastering Oscillations

Life is full of patterns that repeat. From the swing of a grandfather clock to the vibration of a guitar string and the rhythmic pumping of your heart, Oscillations are everywhere.

In this chapter, we move beyond constant velocity and look at Restoring Forces. The big secret? Almost every stable system in the universe, when pushed slightly, will oscillate in a very specific way called Simple Harmonic Motion (SHM).

The Core Pillars of Oscillations

1. Simple Harmonic Motion (SHM)

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

• The Rule: Acceleration a = -ω²x
• The Result: The object doesn’t just return to the center; its momentum carries it past the center, creating a cycle.

2. The Energy Exchange

In a frictionless oscillation, energy is never lost—it just changes form.

• At Extreme positions: Potential Energy is maximum, Kinetic Energy is zero.
• At Mean position (center): Kinetic Energy is maximum, Potential Energy is zero.
• The total energy remains constant throughout the path.

3. The Spring-Mass System

A mass m on a spring with constant k is the “Hello World” of oscillations.

• Time Period (T) = 2π √(m/k)Notice that the time period does not depend on the amplitude. Whether you pull it 1 cm or 5 cm, it takes the same time to complete a cycle!

4. Damped and Forced Oscillations

Real-world oscillations eventually stop due to friction (Damping). If we want them to keep going, we must apply an external force. When the frequency of this external force matches the natural frequency of the system, we get Resonance—massive vibrations that can either make a musical instrument sing or collapse a bridge.

The Gauntlet: 10 Challenging Aptitude Questions

Question 1: The Phase Shift

Two particles are performing SHM of the same amplitude and frequency along the same line. They pass each other while moving in opposite directions when their displacement is half the amplitude. What is the phase difference between them?

Question 2: The Lift Pendulum

A simple pendulum is hanging in a lift. If the lift starts accelerating upward with acceleration a, what happens to the time period? What if the lift is in “Free Fall”?

Question 3: The Spring Cut

A spring of constant k is cut into two equal halves. What is the spring constant of each half? If the original spring had a time period T with mass m, what is the new time period with one of the halves?

Question 4: Velocity vs. Displacement

In SHM, at what displacement from the mean position is the Kinetic Energy equal to the Potential Energy?

Question 5: The Earth Tunnel (Oscillation Edition)

If a hole is bored through the center of the Earth and a ball is dropped, it performs SHM. Calculate the time period of this oscillation. (Take Earth’s radius as 6400 km).

Question 6: The Two-Spring Combo

A mass m is connected between two springs of constants k₁ and k₂.

1. If they are in Series, what is the effective k?
2. If they are in Parallel, what is the effective k?

Question 7: The Seconds Pendulum

A “Seconds Pendulum” is one that takes exactly 1 second to go from one extreme to the other (Time period = 2s). What is the approximate length of such a pendulum on Earth?

Question 8: Superposition of SHM

A particle is subjected to two perpendicular SHMs: x = A sin(ωt) and y = A cos(ωt). What is the path traced by the particle?

Question 9: The Loaded Floating Cylinder

A uniform cylinder of mass M and area A floats vertically in a liquid of density ρ. If it is pushed down slightly and released, it oscillates. Find the time period.

Question 10: Maximum Velocity and Acceleration

A particle performs SHM with amplitude A and angular frequency ω. Find the ratio of its maximum acceleration to its maximum velocity.

Detailed Explanations & Solutions

1. Phase Difference

Using the equation x = A sin(Φ). If x = A/2, then sin(Φ) = 1/2, so Φ = 30° or 150°. Since they move in opposite directions, one is at 30° and the other at 150°.

Result: Phase difference = 150° – 30° = 120° (or 2π/3).

2. Lift Acceleration

Effective gravity g’ = g + a. Since T = 2π√(L/g’), as g’ increases, T decreases.

Result: In free fall (a = g), g’ = 0, so T becomes infinite (the pendulum won’t swing).

3. Spring Cut

Spring constant is inversely proportional to length (k ∝ 1/L). If length is halved, k doubles.

Result: New k = 2k. New Time Period = T/√2.

4. Energy Equality

KE = PE → ½k(A² – x²) = ½kx².

A² – x² = x² → 2x² = A².

Result: x = A / √2 (approx 70.7% of amplitude).

5. Earth Tunnel

The restoring force inside Earth is F = -(GmM/R³)r. This is SHM where the “k” is mg/R.

Result: T = 2π√(R/g) ≈ 84.6 minutes.

6. Spring Combinations

• Series: 1/k_eq = 1/k₁ + 1/k₂ (Softer)
• Parallel: k_eq = k₁ + k₂ (Stiffer)Result: Parallel oscillates faster than Series.

7. Seconds Pendulum

T = 2s. 2 = 2π√(L/9.8).

L = 9.8 / π² ≈ 9.8 / 9.87.

Result: Approximately 1 meter (0.99m).

8. Superposition

x² + y² = A² sin²(ωt) + A² cos²(ωt) = A²(1).

Result: The path is a Circle.

9. Floating Cylinder

The restoring force is the extra buoyant force: F = -(Area × extra_depth × ρ)g.

This matches the spring-like force F = -kx where k = Aρg.

Result: T = 2π √(M / Aρg).

10. Ratio of Maxima

Max Velocity = Aω. Max Acceleration = Aω².

Result: Ratio (Acc/Vel) = ω.