Question

Two particles P and Q describe simple harmonic motions of same amplitude a, frequency v along the same straight line. The maximum distance between the two particles is a`sqrt(2)`. The initial phase difference between the particles is

पर्याय

• zero

• 45°

• 60°

• 90°

Answer 1

90°

Explanation:

The S.H.M. equations for the two particles P and Q should be

x₁ = a sin ωt and x₂ = a sin (ωt + Φ)

∴ x₂ – x₁ = a [sin ωt + Φ) – sin ωt] 

∴ x₂ – x₁ = `a[2cos ((2ωt + phi)/2) cos (phi/2)]`

∴ x₂ – x₁ = `[2a cos(phi/2)][cos(ωt + phi/2)]`

The combined motion of the two particles is described by this expression with changing amplitude.

a' = `2a cos  phi/2`

Given: a' = a`sqrt(2)`

∴ a`sqrt(2) = 2a cos  phi/2`

∴ `cos  phi/2 = 1/sqrt(2)` ⇒ `phi/2` = 45°

∴ Φ = 90°