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#cosine motion

2 public questions tagged with this topic.

Equations of SHM, Phase and Angular Frequency

A particle’s x-projection from circular motion is \( x = 10 \cos (2\pi t) \) (in m). What is its maximum acceleration?

**General equation of SHM** x = A sin(ωt + φ) or A cos(ωt + φ) includes amplitude A (m), angular frequency ω = √(k/m) (rad/s) for spring system, and initial phase φ (rad) setting t=0 position. Phase (ωt + φ) determines instantaneous state, phase difference Δφ governs interference of two SHM motions. Maximum acceleration: aₘₐₓ = ω² A . A = 10 m, ω = 2π s⁻¹ . aₘₐₓ = (2π)² × 10 ≈ 39.48 × 10 ≈ 394.8 m/s² . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 394.8

Ref: NCERT > Physics Book > Oscillations > Equations of SHM, Phase and Angular Frequency

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Energy in SHM - Kinetic, Potential and Total

A particle in SHM has \( x = 3 \cos (5t) \) (in m). What is its kinetic energy at \( x = 1.5 \, \text{m} \) if \( m = 2

**Conservation of mechanical energy** in undamped SHM implies total energy proportional to amplitude squared A² and spring constant k. E = ½ k A² allows amplitude determination from known E and k via A = √(2E/k), with k = m ω² linking dynamical and energetic descriptions for spring-mass system. Total energy: E = (1/2) m ω² A² = 0.5 × 2 × 5² × 3² = 225 J . Potential energy: U = (1/2) m ω² x² = 0.5 × 2 × 25 × (1.5)² = 56.25 J . Kinetic energy: K = E - U = 225 - 56.25 = 168.75 J . Applying

Ref: NCERT > Physics Book > Oscillations > Energy in SHM - Kinetic, Potential and Total

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