SHM POSITION VELOCITY ACCELERATION

Simple harmonic motion carries three kinematic descriptions simultaneously — position, velocity, and acceleration — each a sinusoid at the same frequency but phase-shifted by 90° from the previous. Starting from x(t) = A cos(ωt + φ), one differentiation gives v(t) = −Aω sin(ωt + φ), and a second gives a(t) = −Aω² cos(ωt + φ) = −ω²x. That last equality is Newton's second law: F = ma = −mω²x = −kx. At t = 0.8 s, the phase angle is ωt + φ = 4 × 0.8 + π/3 ≈ 4.247 rad. cos(4.247) ≈ −0.738, sin(4.247) ≈ −0.675. So x = 0.2 × (−0.738) ≈ −0.148 m, v = −0.2 × 4 × (−0.675) ≈ +0.540 m/s, a = −0.2 × 16 × (−0.738) ≈ +2.36 m/s². Energy: k = mω² = 0.5 × 16 = 8 N/m, so E = ½ × 8 × 0.04 = 0.16 J. At this instant, KE = ½ × 0.5 × 0.540² ≈ 0.073 J and PE = ½ × 8 × 0.148² ≈ 0.088 J. Their sum is ≈ 0.161 J — rounding precision away, this agrees with E = 0.16 J. The small discrepancy is purely from rounding the phase; the exact algebra always gives KE + PE = ½kA². That invariance — energy shared between kinetic and potential but always summing to the same total — is what makes SHM a conservative system.