Question

The instantaneous displacement `x` of a particle executing simple harmonic motion is given by `x=a_1sinomegat+a_2cos(omegat+(pi)/(6))`. The amplitude `A` of oscillation is given by
A. `sqrt(a_1^2+a_2^2+2a_1a_2cos((pi)/(6)))`
B. `sqrt(a_1^2+a_2^2+2a_1a_2cos((pi)/(3)))`
C. `sqrt(a_1^2+a_2^2-2a_1a_2cos((pi)/(6)))`
D. `sqrt(a_1^2+a_2^2-2a_1a_2cos((pi)/(3)))`

Answer 1

Correct Answer - D

`x=a_1sinomegat+a_2cos(omegat+(pi)/(6))`
`=a_1sinomegat+a_2sin(omegat+(pi)/(6)+(pi)/(2))`
`=a_1sinomegat+a_2sin(omegat+(2pi)/(3))`
`A^2=a_1^2+a_2^2+2a_1a_2cos((2pi)/(3))`
`a_1^2+a_2^2-2a_1a_2cos((pi)/(3))`
`A=sqrt(a_1^2+a_2^2-2a_1a_2cos((pi)/(3)))`