In the figure, dots and arrows show the position and the velocity of a paritcle executing SHM. What are the phases at the five indicated instants when the position at time t is given by
`x = Asin (omega t + phi)`

Explanation

When `x = A sin ( omega t + phi)`
then`v = A omega cos ( omega t + phi)`
(i) `x=0` , so `sin ( omega t + phi) = 0`, hence `cos ( omega t + phi)= +-1`
`:. V = +- A omega`
Since the arrow is directed towards right, `v= + A omega`
So,` cos ( omega t + phi ) = +1 implies delta = omega t + phi =2 n pi`
`sin ( omega t + phi) = 0 implies delta = npi`
Hence both will be satisfies when `delta =2n pi`
when` n =0, +-1, +-2, +-3....."`
(ii) ` x= +A implies sin ( omega t + phi) = 1 implies dleta = omega t = phi =2n pi +(pi)/(2)`
`v=0 implies cos ( omegat + phi) = 0 implies delta = 2n pi +- (pi)/(2)`
combining these two two, we get ` delta =2npi+ (pi)/(2)`
(iii) `x=0 implies sin ( omega t + pi) = 0 implies delta = omega t + pi = n pi`
`v = -A omega implies cos ( omega t + phi) -1 implies delta = omegat + phi =2 n pi + pi`
Combining these two general values of `delta` , we get
`delta = ( 2n + 1) pi = 2 n pi-pi`
(iv) ` x= - A implies sin ( omega t + phi) = -1 implies delta = 2 n pi + ( 3pi)/(2)`
`v= 0 implies cos ( omega t + phi) = 0 implies delta = 2 n pi + (pi)/(2) ` or ` 2 n pi + (3pi)/(2)`
` :. delta = 2n pi + ( 3pi)/(2)`
(v)` x=0 , v= +A omega implies delta = omega t + phi = 2 n pi + 2pi`
Note `:` `delta_((ii)) - delta_((i))= delta_((iii) )-delta_((ii))= delta_((iv))-delta_((iii))= delta_((v))-delta_((iv))= (pi)/(2)`
If `delta_((i))=0,delta_((ii))=(pi)/(2),delta_((iii))=pi,delta_((iv))= (3pi)/(2), delta_((v))= 2pi`