(a) A particle is subjected to two simple harmonic motions
`x_1=A_1sin omegat` and `x_2=A_2 sin (omegat+pi//3)`.
Find (i) the displacement at `t=0`, (ii) the maximum speed of the particle and (iii) the maximum acceleration of the particle.
(b) A particle is subjected to two simple harmonic motions, one along the x-axis and the other on a line making an angle of `45^@` with the x-axis. The two motions are given by
`xy=x_0sinomegat` and `s=s_0sin omegat`
Find the amplitude of the resultant motion.

Explanation

(a) `x_1=A_1sinomegat`, `x_2=A_2sin(omegat+pi//3)`
`x=x_1+x_2=Asin(omegat+phi)`

`A=sqrt(A_1^2+A_2^2+2A_1A_2cos(phi=pi//3))`
`=(A_1^2+A_2^2+A_1A_2)^(1/2)`
`tan theta=(A_2sinpi//3)/(A_1+A_2cospi/3)=(A_2xxsqrt3//2)/(A_1+A_2xx1//2)`
`=(sqrt3A_2)/(2A_1+A_2)`
(i) `x=Asin(omegat+theta)`
At `t=0`, `x=Asintheta`
`=(A_1^2+A_2^2+A_1A_2)^(1/2)(sqrtA_2)/((4A_1^2+A_2^2+4A_1A_2+3A_2^2)^(1/2))`
`=(sqrt3A_2)/(2)`
OR
`t=0`, `x_1=A_1sinomegat=0`,
`x_2=A_2sin(omegat+pi//3)=(sqrt3A_2)/(2)`
`x=x_1+x_2=(A_2sqrt3)/(2)`
(ii) `v_(max)=omegaA=omega(A_1^2+A_2^2+A_1A_2)^(1/2)`
(iii) `a_(max)=omega^2A=omega^2(A_1^2+A_2^2+A_1A_2)^(1/2)`
(b) `x=x_0 sin omega t` along x-axis
`s=s_0 sin omegat` along `y=x` line
component of s along coordinate axes
`x^'=s_0sinomegatcos45^@=s_0/sqrt2 omegat`
`y^'=s_0sin omegat sin 45^@=s_0/sqrt2 sin omegat`
`X=x+x^'=(x_0+s_0/sqrt2)sin omegat`, `Y=y^'=(s_0)/(sqrt2)sinomegat`
`r=sqrt(X^2+Y^2)=[(x_0+s_0/sqrt2)^2+(s_0/sqrt2)^2]^(1/2)sin omegat`
Resultant amplitude
`=[(x_0+s_0/sqrt2)^2+(s_0/sqrt2)^2]^(1/2)=[x_0^2+s_0^2+sqrt2x_0s_0]^(1/2)`