Oscillations and Simple Harmonic Motion

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Periodic motion: Repeats at regular intervals. Time period (T), Frequency (f = 1/T), Angular frequency (ω = 2πf).
SHM: Restoring force proportional to displacement. F = -kx. Examples: pendulum (small angle), spring-mass.
Pendulum: T = 2π√(L/g). Spring: T = 2π√(m/k).

Oscillations are everywhere — pendulum clocks, sound, vibrations. SHM is the simplest oscillation. NDA tests time period formulas and resonance.

Periodic and Oscillatory Motion

Periodic motion: Repeats at equal intervals. Example: Earth's revolution around Sun.
Oscillatory motion: Periodic motion about a fixed equilibrium position. Example: pendulum, spring.
Time period (T): Time for one complete cycle.
Frequency (f or ν): Cycles per second. f = 1/T. Unit: Hz.
Angular frequency (ω): ω = 2πf = 2π/T.
Amplitude (A): Maximum displacement from equilibrium.

Simple Harmonic Motion

Definition: Motion in which restoring force is directly proportional to displacement and opposite in direction.
F = -kx (Hooke's law form).
x(t) = A·sin(ωt + φ) — displacement as function of time.
Velocity v = Aω·cos(ωt + φ); max at equilibrium, zero at extremes.
Acceleration a = -ω²x; max at extremes, zero at equilibrium.
Energy: Total = KE + PE = ½kA² (constant). At equilibrium: KE = max, PE = 0. At extremes: KE = 0, PE = max.

Simple Pendulum and Spring

Simple pendulum (small angle):
- Period: T = 2π√(L/g).
- L = length, g = acceleration due to gravity.
- Period independent of mass and (for small angles) amplitude.
- Seconds pendulum: T = 2 s. Length L ≈ 99.4 cm.
Spring-mass system:
- Period: T = 2π√(m/k).
- k = spring constant.
- Period independent of g (works in space).

Damping and Resonance

Damped oscillations: Real systems lose energy to friction/air. Amplitude decreases over time.
Forced oscillations: External periodic force drives the system.
Resonance: When driving frequency = natural frequency, amplitude grows large.
Examples: Pushing a swing in time; opera singer breaking glass; Tacoma Narrows bridge collapse (1940) — wind-driven resonance.
Engineering use: Tune radios to resonant frequency. Avoid resonance in machinery (bridges, buildings).

NDA PYQ Examples

Q: The period of a simple pendulum depends on:

(a) Length and mass (b) Length and g only (c) Length only (d) Mass and g

Answer: (b) Length and g only — independent of mass.

Q: At the mean position of SHM:

(a) Velocity is zero (b) Acceleration is zero (c) KE is zero (d) PE is maximum

Answer: (b) Acceleration is zero (at equilibrium, no restoring force needed).

Q: A pendulum clock will run slow:

(a) On a hill top (b) At the equator (c) In a deep mine (d) All of the above

Answer: (d) All — g decreases with altitude, equator (centrifugal), depth. T = 2π√(L/g), so lower g = larger T = slower clock.

Q: What is the frequency of a seconds pendulum?

(a) 0.5 Hz (b) 1 Hz (c) 2 Hz (d) 60 Hz

Answer: (a) 0.5 Hz — T = 2 s, f = 1/T.

Drill Oscillations and Simple Harmonic Motion for NDA

NDA-pattern items on Oscillations and Simple Harmonic Motion with answer keys and explanations.

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Frequently Asked Questions

Why does a pendulum clock run slow on a mountain?

Because g decreases with altitude. T = 2π√(L/g); lower g means larger T (longer period), so the clock runs slow. Adjust by shortening the pendulum slightly.

What is resonance?

When a system is driven at its natural frequency, the amplitude grows large. A child on a swing receives small pushes that, properly timed (resonant), produce large swings. Same principle drives radio tuning, musical instruments, even building damage in earthquakes.

Can a spring-mass system oscillate in space?

Yes. The spring's period T = 2π√(m/k) doesn't depend on g. The pendulum, by contrast, needs gravity (T depends on g) and would not work in zero gravity.

Why is amplitude small important for pendulum formula?

T = 2π√(L/g) is exact only for small angles (< 10° approximately). For larger angles, the formula needs correction terms. The approximation works because sin θ ≈ θ for small θ.

What is SHM mathematically?

Motion described by sinusoidal function. x(t) = A·sin(ωt + φ). The defining condition: restoring force is proportional to displacement and opposite in direction. F = -kx → a = -ω²x.