#cosine

Forced Oscillations and Resonance

What underlies the periodic nature of SHM when expressed as a superposition of sine and cosine functions?

**Resonance phenomenon** amplifies response when driving frequency matches natural frequency ω_d ≈ ω₀, large amplitude even with small F₀, as damping limits growth. Natural frequency determined by system parameters, resonance condition crucial for understanding vibrations and energy absorption, e.g., bridge collapse, tuning. The periodicity arises from the repeating nature of sine and cosine functions, which have a fixed period ( 2π/ω ), ensuring the motion repeats consistently. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result The periodicity of trigonome

Ref: NCERT > Physics Book > Oscillations > Forced Oscillations and Resonance

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Damped Oscillations

A mass oscillates with \( v = -8 \cos (2t) \) (in m/s). What is its displacement function?

**Effect of damping** is gradual amplitude reduction while period remains nearly constant for light damping. Mechanical energy decreases as work done against damping force, E(t) = ½ k A(t)² decaying exponentially, and motion ceases without external energy input, distinguishing from ideal undamped SHM. Velocity: v = -ω A sin (ω t) , but given v = -8 cos (2t) . ω = 2 s⁻¹, vₘₐₓ = ω A = 8 ⇒ A = (8/2) = 4 m . Since v = -A ω sin (ω t) , adjust phase: x = 4 sin (2t) . Applying x = A cos(ωt + φ), v = -ωA

Ref: NCERT > Physics Book > Oscillations > Damped Oscillations

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Damped Oscillations

A particle’s x-projection from circular motion is \( x = 6 \cos (3t) \) (in m). What is its maximum acceleration?

**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. Maximum acceleration: aₘₐₓ = ω² A . A = 6 m, ω = 3 s⁻¹ . aₘₐₓ = 3² × 6 = 9 × 6 = 54 m/s² . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 54 m/s² follows, reflecting SHM dependence on amplitude A, ω and system parameters.

Ref: NCERT > Physics Book > Oscillations > Damped Oscillations

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Periodic Motion and SHM Basic Concepts

A particle’s displacement is \( x = 3 \cos (4\pi t + \frac{\pi}{3}) \) (in m). What is its velocity at \( t = 0 \, \text

**Periodic motion** repeats after fixed period T, x(t+T)=x(t), while oscillatory motion involves to-and-fro about equilibrium. SHM is special periodic motion where restoring force proportional to displacement, F = -k x, acceleration a = -ω² x, leading to sinusoidal displacement x = A cos(ωt + φ). Velocity: v = -ω A sin (ω t + Φ) . A = 3 m, ω = 4π s⁻¹, Φ = (π/3) . At t = 0 : v = -4π × 3 sin (π/3) = -12π × (√(3)/2) ≈ -32.58 m/s . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and

Ref: NCERT > Physics Book > Oscillations > Periodic Motion and SHM Basic Concepts

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