Simple Harmonic Motion

The total displacement is
x = x1(t)+x2(t)+x3(t) = 0
A sin ωt + Asin(ωt+

2π

3

) + B sin (ωt + ϕ) = 0
Asinωt(1+cos

2π

3

) + A cos ωt sin

2π

3

+ B sin (ωt + ϕ) = 0

A

2

sinωt+

√3

2

Acosωt = - B sin ωt cos ϕ - B cos ωt sin ϕ
Therefore,

A

2

= - B cos ϕ (1)
and

√3

2

A = - B sin ϕ (2)
Substituting A from Eq. (1), we get
√3cosϕ = - sin ϕ ⇒ ϕ =

4π

3

and by back substituting, we get B = A.