Simple harmonic motion — A/L Combined Mathematics

What must I remember about Simple harmonic motion?

• Definition of Simple Harmonic Motion (SHM): Rectilinear motion of a particle where acceleration is directly proportional to its displacement from a fixed center O and directed ALWAYS toward that center: x'' = -ω² * x (ω > 0 is angular frequency).
• Kinematics differential equation solution: Velocity-displacement relation: v² = ω² * (a² - x²), where a is the Amplitude (maximum displacement from center O where v = 0).
• Maximum speed and acceleration: Maximum speed occurs at center O (x = 0): v_max = a * ω; Maximum acceleration occurs at extremities (x = ±a): a_max = a * ω².
• Displacement as a function of time: x(t) = a * sin(ωt + ε) or a * cos(ωt + φ); if measured from center O at t = 0 (x = 0): x(t) = a * sin(ωt) and v(t) = a*ω * cos(ωt); if measured from extremity at t = 0 (x = a): x(t) = a * cos(ωt).
• Period and Frequency: Period T = 2π / ω (independent of amplitude a — isochronous motion); Frequency f = 1 / T = ω / (2π).

Test yourself (1 minute)

1. Auxiliary Circle representation

   SHM along diameter can be modeled as projection of uniform circular motion of radius a with angular speed ω on the auxiliary circle; displacement x = a cos θ, phase angle θ = ωt.

2. Energy in SHM

   Kinetic energy E_k = (1/2)m v² = (1/2)m ω² (a² - x²); Potential energy E_p = (1/2)m ω² x²; Total mechanical energy E = E_k + E_p = (1/2)m ω² a² (constant).

3. Time to travel between positions

   Time from center (x = 0) to x = x1 is t = (1/ω) arcsin(x1/a); Time from extremity (x = a) to x = x1 is t = (1/ω) arccos(x1/a).

Practise Simple harmonic motion questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Applied Mathematics, Unit 6

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Year 2 · Term 3