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Question

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

A
60
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B
30
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C
90
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D
120
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Solution

The correct option is D 120
For a particle executing SHM, we can say that
x=Asin(ωt+ϕ)
Given two particles are executing SHM, we can write that
x1=Asin(δ1) and x2=Asin(δ2)
Where δ1 and δ2 are the phase of two particles.
From the data given in the question, two particles pass one another going in opposite directions when their displacements are half of their amplitudes.
x1=x2=A2, we get
For particle 1:
Assuming particle is moving away from mean position.
A2=Asin(δ1)sin(δ1)=12δ1=π6
Similarly, For particle 2:
Assuming particle is moving towards mean position.
A2=Asin(δ2)sin(δ2)=12δ2=5π6

Phase difference between the two particles is δ=|δ1δ2|=5π6π6=2π3 or 120
Thus, option (d) is the correct answer.

Alternate solution :
Phasor diagram of the particles executing SHM is shown below.


From the diagram we can deduce that,cosϕ=xA
When the particles are crossing each other at x=A2 we get,
cosϕ=A2Aϕ=cos112ϕ=π3
so, phase difference between the two particles is given by,
δ=2ϕ=2π3 or 120
Thus, option (d) is the correct answer.

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