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Question

A particle starts from mean position x=0 and starts moving towards negative extreme position. Find the initial phase constant ϕ of the particle.

A
ϕ=0
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B
ϕ=π2
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C
ϕ=π
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D
ϕ=π4
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Solution

The correct option is C ϕ=π

Since the particle starts at mean position,
we know that, the displacement can be written as
x=A sin(ωt+ϕ) (1)
From this by differentiating we get,
V=A ω cos(ωt+ϕ) (2)
Since, particle was initially at x=0, from equation 1 we get,
0=A sin ϕsin ϕ=0ϕ=nπ where n=0,1,2,3.....
ϕ can be 0 , π , 2π.....
Since the particle is moving towards negative extreme position, at t=0, Particle velocity is negative
A ω cos ϕ<0
cos ϕ<0
If, ϕ=0,2π,4π... then cos ϕ>0
But, If ϕ=π,3π,5π.... then cos ϕ<0
So, we can conclude that ϕ=π,3π,5π.....
Hence option (c) is the correct answer.

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