The equation of motion of a particle executing SHM is `x=Asin(omegat+theta)`. Calculate the velocity and acceleration of the particle. If m is the mass of the particle, then what is the maximum value of its kinetic energy?

Explanation

Given equation of SHM, `x=Asin(omegat+theta)" "...(1)`
So, velocity, v = `(dx)/(dt)=omegaAcos(omegat+theta)`
From equation (1) we get,
`sin(omegat+theta)=x/A`
`therefore" "cos(omegat+theta)=sqrt(1-sin^(2)(omegat+theta))=sqrt(1-x^(2)/A^(2))=sqrt((A^(2)-x^(2))/A^(2))`
`therefore" "v=omegaAsqrt((A^(2)-x^(2))/A^(2))=omegasqrt(A^(2)-x^(2))`
Again, acceleration of the particle,
`a=(dv)/(dt)=d/(dt){omegaAcos(omegat+theta)}=-omega^(2)Asin(omegat+theta)`
`=-omega^(2)x" "[becausex=Asin(omegat+theta)]`
As the maximum value of `cos(omegat+theta)` is 1,
the maximum value of velocity, `v_(max)=omegaA`
`therefore" "` Maximum kinetic energy of the particle
`=1/2mv_(max)^(2)=1/2momega^(2)A^(2)`.