Dynamical quantities associated with SHM

Displacement

• The displacement of a particle executing simple harmonic motion is

           x = a sin (ω₀t + θ)              − 1 < sin (ω₀t + θ) < 1

           x_max = ± a                            (Extreme position)

           x_min = 0                                (Mean position)

           | x_max | = a                           (Amplitude of particle)

Velocity

           At extreme position, x = ± a  ⇒  v_min = 0

           At mean position, x = 0         ⇒  v_max = ± aω₀

Acceleration

           At extreme position, x = ± a  ⇒  f = | f_max | = aω₀²

           At mean position, x = 0          ⇒  f = f_min = 0

Phase

• The angle of different physical quantities (displacement, velocity and acceleration) of S.H.M. are called its phase.
• If t = 0, then ቀ₀ = θ (initial phase or epoch)

Time period

• The time taken by particle to complete one oscillation is known as time period.

Here   x (t) = x (t + T)                         [∵ a sin (ω₀t + θ)]

∴       a sin (ω₀t + θ) = a sin [(ω₀(t + T) + θ)]

or      ω₀t + θ + 2nπ = ω₀(t + T) + θ

or      2nπ = ω₀T    ⇒     T = 2nπ / ω₀

∵       n = n_min = 1

∴       T = 2π / ω₀

or     T = 2π √ (m / k)                    [∵   ω₀ = √ (k / m)]

Frequency

• The reciprocal of time period is known as frequency (ν).