Write the equation of SHM for the situations show below:

Explanation

a. We know general equation of SHM is
`x = A sin (omega t + phi)`
At `t = 0, x = + A`
`A = A sin (phi) implies sin phi = 1: Hence, phi = pi//2`
`x = A sin (omega + (pi)/(2)) = A cos omega t`
b. `x = A sin (omega t + phi)`
At `t = 0, x = - A`
`-A = A sin (phi) implies cos phi = 1: Hence, phi = (3 pi)/(2)`
`x = A sin (omega + (3 pi)/(2))`
`x = - A cos omega t`
c. `x = A sin (omega t + phi)`
At `t = 0, x = (A)/(2)`
`(A)/(2) = A sin (phi) implies sin phi = (1)/(2), Hence, phi = (pi)/(6), (5 pi)/(6)`
The partical is moving towards the mean position and in negative direction , i.e. velocity is negative . The value of `phi` should be selected such that `cos phi` should be negative.
`Velocity v = A omega cos (omega t + phi)`
At `t = 0, v = - ve`
`v = A omega cos phi`
Hence `phi = (5 pi)/(6)`
`x = A sin (omega t + (5 pi)/(6))`