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Find the displacement equation of the si...

Find the displacement equation of the simple harmonic motion obtained by combining the motion.
`x_(1) = 2sin omega t`, `x_(2) = 4sin (omega t + (pi)/(6))` and `x_(3) = 6sin (omega t + (pi)/(3))`

Text Solution

Verified by Experts

The correct Answer is:
D
The resultant equation is,
`x = A sin (omega t + phi)`
`Sigma A_(x) = 2 + 4 cos 30^(@) + 6cos 60^(@) =8.46`
and `Sigma A_(y) = 4sin 30^(@) + 6cos 30^(@) = 7.2`
`:. A = sqrt((Sigma A_(x))^(2) + (Sigma A_(y))^(2))`
`= sqrt((8.46)^(2) + (7.2)^(2))`
` = 11.25`
and `tan phi = (Sigma A _(y))/(Sigma A_(x))`
`= (7.2)/(8.46) = 0.85`
or `phi =tan^(-1)(0.85)`
`= 40.4^(@)`
Thus, the displacement equation of the combined motion is,
`x = 11.25sin (omega t + phi)`
where, `phi = 40.4^(@)`
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