(a) A particle is subjected to two SHMs of same time period in the same direction. If `A_1=6cm`, `A_2=8cm`, find the resultant amplitude if the phase difference between the motions is (a) `0^@`, (b) `60^@`, (c) `90^@` and (d) `120^@`.
(b) In the previous problem if `A_1=A_2=a` and resultant amplitude is also a, find phase difference between them.
(c) Three SHMs of equal amplitude A and equal time periods in the same direction combine. The phase of the second motion is `120^@` ahead of first and the phase of the third motion is `120^@` ahead of the second. Find the amplitude of the resultant motion.

Explanation

(a) `x_1=A_1sinomegat`
`x_2=A_2sin(omegat+phi)`
The resultant motion
`x=x_1+x_2=Asin(omegat+theta)`
where `A=sqrt(A_1^2+A_2^2+2A_1A_2cosphi)`
(i) `phi=0^@`, `A=A_1+A_2=6+8=14cm`
(ii) `phi=60^@`, `A=sqrt((6)^2+(8)^2+2xx6xx8xxcos60^@`
`=sqrt(148)cm`
(iii) `phi=90^@`, `A=sqrt((6)^2+(8)^2)=10cm`.
(iv) `phi=120^@`, `A=sqrt((6)^2+(8)^2+2xx6xx8xxcos120^@)`
`=sqrt(52)cm`.
(b) `A_1=A_2=A=a`
`A^2=A_1^2+A_2^2+2A_1A_2cosphi`
`a^2=a^2+a^2+2a^2cosphi`
`cos phi=-1/2impliesphi=120^@`
(c) `x_1=Asinomegat`, `x_2=Asin(omegat+120^@)`,
`x_3=Asin(omegat+240^@)`
`x=x_1+x_2+x_3=A_Rsin(omegat+theta)`

Resultant amplitude `A_g=0`