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Find the displacement equation of the si...

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

Text Solution

Verified by Experts

The resultant equation is ,
`x=A "sin"(omegat+phi)`
`sumA_(x)=2+4 "cos" 30^(@)+6 "cos"60^(@)=8.46`

and `sumA_(y)=4 "sin"30^(@)+6 "sin"60^(@)=7.2`
`therefore A=sqrt((sumA_(x))^(2)+(sumA_(y))^(2))`
`=sqrt((8.46)^(2)+(7.2)^(2))=11.25`
and `tanphi=(sumA_(y))/(sumA_(x))=(7.2)/(8.46)=085`
or `phi=tan^(-1)(0.85)=40.4^(@)`
Thus, the displacement equation of the combined motion is,
`x=A"sin"(omegat+phi)`
`x=11.25 "sin"(omegat+phi)`
where `phi=40.4^(@)`
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