Oscillations & Simple Harmonic Motion

• Formulate and solve the second-order differential equation of SHM: d^2x/dt^2 + ω^2 x = 0.
• Derive expressions for displacement, velocity, acceleration, and total energy in SHM.
• Calculate time period of simple pendulum, compound pendulum, and loaded springs.
• Distinguish between free, damped, and forced oscillations and resonance.

• Definition of SHM (Restoring force F = -kx)
• Displacement Equation: x(t) = A sin(ωt + φ)
• Velocity v = ω sqrt(A^2 - x^2) & Acceleration a = -ω^2 x
• Total Energy Conservation (E = 1/2 m ω^2 A^2)
• Simple Pendulum Period: T = 2π sqrt(L/g)
• Spring Combinations (Series & Parallel)

“SHM is the projection of uniform circular motion along any diameter.”

• •Maximum velocity occurs at mean position (x = 0), zero at extremes.
• •Maximum acceleration occurs at extreme positions, zero at mean.
• •Average kinetic and potential energy over one period are each equal to E/2.

High-density problems covering spring network periods, energy partitions in SHM, and pendulum mechanics.