Problem Solving Simple Harmonic Motion
Concepts to remember on ProblemSolving Simple Harmonic Motion for JEE and CBSE board exams:
Simple Harmonic Motion (SHM):
 Periodic motion where the restoring force is directly proportional to the negative displacement.
 Examples: Pendulum, springmass system, and oscillating bodies.
 Motion takes place along a straight line.
Characteristics of SHM:

Displacement (x): A sinusoidal function of time (t) with amplitude (A) and angular frequency (ω).

x(t) = A cos(ωt + Φ) or x(t) = A sin(ωt + Φ)

Velocity (v): Also sinusoidal, 90° out of phase with displacement.

v(t) = ωA sin(ωt + Φ) or v(t) = ωA cos(ωt + Φ)

Acceleration (a): Sinusoidal, 180° out of phase with displacement and proportional to (Aω²).

a(t) = ω²A cos(ωt + Φ) or a(t) = ω²A sin(ωt + Φ)
Equations of SHM:
 Displacement: x = A cos(ωt + ∅), where ∅ is the phase constant.
 Velocity: v = Aω sin(ωt + ∅).
 Acceleration: a = Aω² cos(ωt + ∅).
Time Period (T):
 Time taken for one complete oscillation.
 T = 2π√(m/k), where m is the mass of the oscillating object, and k is the spring constant.
Frequency (f):
 Number of oscillations per second.
 f = 1/T = ω/2π.
Angular Frequency (ω):
 Measures the rate of change of phase angle.
 ω = 2πf = √(k/m).
Phase:
 Describes the position of an oscillating particle within its cycle.
 Phase difference: Describes the difference in phase between two oscillations.
Energy in SHM:
 Total energy (E) is constant and the sum of kinetic energy (K) and potential energy (U).
 K = ½kA², and U = ½kA² cos²(ωt + ∅).
 E = ½kA².
Resonance:
 Occurs when the frequency of an applied periodic force matches the natural frequency of the system.
 Causes maximum amplitude and energy transfer.