A point participates simultaneously in two harmonic oscillations of the same direction`:x_(1) =a cos omega t ` and `x_(2)=a cos 2 omega t. ` Find the maximum velocity of the point .

Explanation

Given,` x_(1)= a cos omegat ` and `x_(2)= a cos 2 omegat `
so, the net displacement,
`x=x_(1)+x_(2)=a{cos omegat +cos 2 omegat}=a{cos omegat+2 cos ^(2) omegat-1}` and `v_(x)=x=a{-omega sin omegat -4 omega cos omegat sin omega t }`
For `x` to be maximum,
` ddot x=a omega^(2)cos omegat -4 a omega^(2)cos^(2)omegat+4 a omega^(2)sin ^(2)omegat=0`
or, `8 cos^(2)omegat+cos omegat-4=0,` which is a quadratic equation for `cos omegat`.
Solving for accepctable value
` cos omegat =0.644`
thus `sin omegat=0.765`
and `v_(max)=|v_(x_(max))|=+a omega[0.765=4xx0.765xx0.644]=+2.73 a omega`