Two waves executing simple harmonic motion travelling in the same direction with same amplitude and frequency are superimposed. The resultant amplitude is equal to the √3 times of amplitude of individual motions. The phase difference between the two motions is (degree)

Question

Two waves executing simple harmonic motion travelling in the same direction with same amplitude and frequency are superimposed. The resultant amplitude is equal to the √3 times of amplitude of individual motions. The phase difference between the two motions is (degree) (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

A text resultant =√ A 12+ A 22+2 A 1 A 2 cos φ ⇒ √3 A =√ A 2+ A 2+2 A 2 cos φ ⇒ 3 A 2=2 A 2+2 A 2 cos φ ⇒ cos φ=(1/2) ∴ φ=60° ∴ Phase difference =60 degree

Q. Two waves executing simple harmonic motion travelling in the same direction with same amplitude and frequency are superimposed. The resultant amplitude is equal to the $3$ times of amplitude of individual motions. The phase difference between the two motions is _______(degree)

$A_{resultant} = A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} cos ϕ$
$\Rightarrow 3 A = A^{2} + A^{2} + 2 A^{2} cos ϕ$
$\Rightarrow 3 A^{2} = 2 A^{2} + 2 A^{2} cos ϕ$
$\Rightarrow cos ϕ = \frac{1}{2}$
$∴ ϕ = 6 0^{\circ}$
$∴$ Phase difference $= 60$ degree

Q. Two waves executing simple harmonic motion travelling in the same direction with same amplitude and frequency are superimposed. The resultant amplitude is equal to the 3​ times of amplitude of individual motions. The phase difference between the two motions is _______(degree)

Aresultant ​=A12​+A22​+2A1​A2​cosϕ​ ⇒3​A=A2+A2+2A2cosϕ​ ⇒3A2=2A2+2A2cosϕ ⇒cosϕ=21​ ∴ϕ=60∘ ∴ Phase difference =60 degree

Q. Two waves executing simple harmonic motion travelling in the same direction with same amplitude and frequency are superimposed. The resultant amplitude is equal to the $3$ times of amplitude of individual motions. The phase difference between the two motions is _______(degree)

$A_{resultant} = A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} cos ϕ$
$\Rightarrow 3 A = A^{2} + A^{2} + 2 A^{2} cos ϕ$
$\Rightarrow 3 A^{2} = 2 A^{2} + 2 A^{2} cos ϕ$
$\Rightarrow cos ϕ = \frac{1}{2}$
$∴ ϕ = 6 0^{\circ}$
$∴$ Phase difference $= 60$ degree