Problem Solving Simple Harmonic Motion

Concepts to remember on Problem-Solving 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, spring-mass 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.