Find an expression for the velocity and acceleration of a particla executing SHM.

Explanation

Displacement of a particle executing SHM is given b `y=asin omegat` .. . (i)
Differentiating both the sides of equation (i) w.r.t., .t. we get,
`(dy)/(dt)=(d)/(dt)(a sin omegat)=A(d)/(dt)sinomega`
`v=A cos omegat.omega=A omega cos omegat` . . . (ii)
As `sin^(2)omegat+cos^(2)omegat=1`
`rArrcos^(2)omegat=1-sin^(-1)sin^(2)omegat" or "cosomegat=sqrt(1-sin^(2)omegat)` . . .(iii)
From equation (i), `sinomegat=(y)/(A)` . . .(iv)
Substituting (iv) in (iii), we get,
`cos omegat=sqrt(1-(y^(2))/(A^(2)))=sqrt((A^(2)-y^(2))/(A^(2)))" or "cos omegat=(sqrt(A^(2)-y^(2)))/(A)` . .. (v)
Substituting equation (v) in (ii), we get `v=Aomega(sqrt(A^(2)-y^(2)))/(A)`
Velocity, `v=omegasqrt(A^(2)-y^(2))` . . . (vi)
Differentiating (ii), we get
`(dv)/(dt)=Aomega(d)/(dt)cosomegat=-Aomegasinomegat. omega `
`a=-Aomega^(2)sinomegat` . . .(vii)
Substituting equation (i) in (vii), we get
`a=-omega^(2)y`