Correct Answer - Option 2 : 90°
CONCEPT:
-
Simple harmonic motion: The motion in which the restoring force is directly proportional to the displacement from equilibrium is simple harmonic motion.
⇒ F α -x
Where F = force and x = the displacement from mean position equilibrium.
- The equation of displacement in Simple Harmonic Motion is given by:
⇒ x = A sin(ωt + ϕ) .........(i)
where x is the distance from the mean position or equilibrium at any time t, A is amplitude (max displacement), ω is the angular frequency, and t is time.
- The equation of velocity in Simple Harmonic Motion is given by differentiating equation (i)
\(⇒ v={dx \over dt} = {d (A sin(ωt + ϕ)) \over dt}\)
⇒ v = Aω cos(ωt + ϕ) ...........(ii)
where v is the velocity at any time t, A is amplitude (max displacement), ω is the angular frequency, and t is time.
- The equation of acceleration in Simple Harmonic Motion is given by differentiating equation (ii)
\(⇒ a={dv \over dt} = {d (A\omega cos(ωt + ϕ)) \over dt}\)
⇒ a = -Aω2 sin(ωt + ϕ)
where a is the acceleration at any time t, A is amplitude (max displacement), ω is the angular frequency, and t is time.
CALCULATION:
⇒ v = Aω cos(ωt + ϕ)
⇒ v = Aω sin(ωt + ϕ + π/2)
Phase of displacement = ωt + ϕ + π/2
- Equation of acceleration is
⇒ a = -Aω2 sin(ωt + ϕ)
⇒ a = Aω2 sin(ωt + ϕ + π)
Phase of acceleration = ωt + ϕ + π
- Difference between phase of acceleration and phase of velocity
⇒ Δϕ = (ωt + ϕ + π) - (ωt + ϕ + π/2) = π/2 or 90°
- So velocity lags acceleration by 90°. So the correct answer is option 2.
