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#small-angle approximation

2 public questions tagged with this topic.

Damped Oscillations

For a pendulum executing small oscillations, what approximation allows its motion to be treated as simple harmonic?

**Real oscillators** experience damping, amplitude decreasing with time. Critical damping returns to equilibrium fastest without oscillation, overdamping slows return, underdamping shows decaying oscillations, classification based on b relative to 2mω₀, important for practical systems. For small angles, sin θ ≈ θ (in radians), making the restoring torque ( tau = -mgL θ ) linear, a requirement for SHM. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result sin θ ≈ θ follows, reflecting SHM dependence on amplitude A, ω and system parameters.

Ref: NCERT > Physics Book > Oscillations > Damped Oscillations

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Simple Pendulum and Angular SHM

For a simple pendulum, why does the approximation of SHM break down at large amplitudes?

**Simple pendulum** for small angles approximates SHM with period T = 2π√(L/g), frequency f = (1/2π)√(g/L), angular frequency ω = √(g/L) (rad/s), independent of mass. L length from pivot to centre of mass (m), g = 9.8 m/s² acceleration due to gravity, approximation sinθ ≈ θ (rad) for θ < 10°. At large amplitudes, sin θ neq θ , and higher-order terms in the expansion ( sin θ = θ - (θ³/6) + ldots ) become significant, making the restoring force non-linear. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result

Ref: NCERT > Physics Book > Oscillations > Simple Pendulum and Angular SHM

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