Derive expressions for the period of SHM in terms of (1) angular freq

1.	Derive expressions for the period of SHM in terms of (1) angular frequency (2) force constant (3) acceleration.
Answer» The general expression for the displacement (x) of a particle performing SHM is x = A sin (ωt + α) (1) Let T be the period of the SHM and x the displacement after a further time interval T. Then x₁ = A sin [ω(t + T) + α] = A sin (ωt + ωT + α) = A sin (ωt + α + ωT) Since T ≠ 0, for x₁ to be equal to x, we must have (ωT)_min = 2π. Hence, the period (T) of SHM is T = 2π/ω This is the expression for the period in terms of the constant co, the angular frequency. (2) If m is the mass of the particle and k is the force constant, ω = \(\sqrt{k/m}\). ∴T = \(\frac{2\pi}\omega\) = \(\frac{2\pi}{\sqrt{k/m}}\) = 2π \(\sqrt{\frac mk}\) (3) The acceleration of a particle performing SHM has a magnitude a = ω²x ∴ ω = \(\sqrt{a/x}\) = \(\sqrt{acceleration\,per\,unit\,displacement}\) ∴ T = \(\frac{2\pi}\omega\) = \(\frac{2\pi}{\sqrt{acceleration\,per\,unit\,displacement}}\)

Derive expressions for the period of SHM in terms of (1) angular frequency (2) force constant (3) acceleration.

Answer»

The general expression for the displacement (x) of a particle performing SHM is x = A sin (ωt + α)

(1) Let T be the period of the SHM and x the displacement after a further time interval T. Then

x₁ = A sin [ω(t + T) + α]

= A sin (ωt + ωT + α)

= A sin (ωt + α + ωT)

Since T ≠ 0, for x₁ to be equal to x, we must have (ωT)_min = 2π.

Hence, the period (T) of SHM is T = 2π/ω This is the expression for the period in terms of the constant co, the angular frequency.

(2) If m is the mass of the particle and k is the force constant, ω = \(\sqrt{k/m}\).

∴T = \(\frac{2\pi}\omega\) = \(\frac{2\pi}{\sqrt{k/m}}\) = 2π \(\sqrt{\frac mk}\)

(3) The acceleration of a particle performing SHM has a magnitude a = ω²x

∴ ω = \(\sqrt{a/x}\)

= \(\sqrt{acceleration\,per\,unit\,displacement}\)

∴ T = \(\frac{2\pi}\omega\) = \(\frac{2\pi}{\sqrt{acceleration\,per\,unit\,displacement}}\)