Characteristics of simple harmonic motion

• Displacement: y = asin (ωt + ɸ)
• Velocity v = dy/dt = aω cos(ωt + ɸ)
• Accelaration a = dv/dt = - aω² sin(ωt + ɸ)
• Time period T = 2∏ √ (displacement/ accelartion)
• Frequency f = 1/T = 1/2∏√ accelartion / displacement
• Potential energy U = ½ mω²y²
• Kinetic energy KE = ½ mω²(a² – y²)

 where a = amplitude

          ω = angular frequency

          ɸ = Phase difference

        Springs

• Time period T = 2∏√ (inertia factor / spring factor) = 2∏√ (m /k)
• If the vibrations of the spring are horizontal, frequency f – 1/T = = 1/2∏√ (k /m)
• If the vibrations of the spring are vertical, f – 1/T = = 1/2∏√ (g /l)
• Spring constant of parallel springs k₁ + k₂
• Spring constant of series springs 1/k = 1/k₁ + 1/k₂

         Simple pendulum

• T = 2∏√ (inertia factor / spring factor) = 2∏√ (m /k)
• If the vibrations of the spring are horizontal T = 2∏√ (l /g)