Question

Two particles execute SHM of same amplitude and frequency along same straight line.They pass one another when going in opposite directions.Each time their displacement is half of their amplitude.What is the phase difference between them?​

Answer 1

Since the 2 particles undergo SHM,

their displacements can be given by y=Asin(wt) y = sin(wt + p)

Here for convenience, i used w for omega, and p for phi for phase difference that we need to find.

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Hence,

Y = A Sin(wt) Y = Asin(wt + p)

Here, as given, Y = A/2.

Hence,

A/2 = A Sin (wt) A/2 = Asin(wt+p)

Therefore, Sin(wt) = 1/2

Hence Cos(wt)= root (1-1/4) = (root3) / 2

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On the RHS, we hav, Sin(wt + p) = 1/2.

As we know, Sin(x+y) = SinxCosy + CosxSiny

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Sin(wt+p) = Sin(wt)Cosp + Cos(wt)Sinp

Substitutin values of Sin(wt) and Cos(wt):-

Sin(wt+p) = (1/2)Cosp + (root3 / 2)Sinp = 1/2

= Cosp + (root3)Sinp = 1

(root3)Sinp = 1 -Cosp

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Squaring both sides,

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3Sin2(p) = (1-Cosp)2

3Sin2(p) =1 + Cos2p - 2Cosp

Here, Sin2p= 1 - Cos2p,

Hence,

3(1 - Cos2p) =1 + Cos2p - 2Cosp

3 - 3Cos2p = 1 + Cos2p - 2Cosp

4Cos2p - 2Cosp - 2 = 0

2Cos2p - Cosp - 1 = 0

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Hence, Cosp = { 1 +- root[1 - 4(2)(-1)] } / 2(2)

Cosp = (1 +- 3) / 4

Hence, Cosp = 1or Cosp = -1/2

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Here, Cosp cannot be 1 since p cant be 0 as the particles are moving in opp. directions and so, Cosp = -1/2.

Therefore, p = 120 degrees

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The phase difference between the 2 particles = 120o

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Answer 2

We have x=Asin(ωt+δ); Where δ is the phase constant.

As the particle passes at x=A/2.

Let it happen at t=0.

Alos let the particle 1 and particle 2 move in positive x direction and negative x direction at x=0 respectively.

So for particle 1 velocity should be positive and for particle 2 it should be negative .

For particle 1.

At x=A and t=0 A

A2=Asinδ

⇒1/2=sinδ.

⇒δ =π6,5π6

The velocity is

v=dxdt=Awcos(wt+δ)At t=0, v=AwcosδNow cos(π/6)=3√2 and cos(5π/6)=−3√2

As v is positive for particle 1, δ must be equal to π/6

For particle 2, v is negative , δ must be equal to 5π/6

Thus the phase difference between them is equal to difference in phase constant =4pi/6=120o

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