In the eqaution of shm
x= A sin (wt +¢)
Put x=0, t=0
We get sin(¢)= 0
Which gives solution,
¢= 0,± π, ±2π, ±3π........ And so on so forth
Now put t=0 and ¢= 0, ±π, ± 2π ..... In the velocity eq
V= Aw cos (wt + ¢)
V= Aw, –Aw, Aw, – Aw,..... Respectively.
Which means that when ¢ = even multiple of π then V is in right direction at mean position at t=0
When ¢= odd multiple of π then V is towards left direction at mean position at t=0.
Since we usually assume the beginning of shm of particle from mean position towarss right therefore ¢= even multiple of π (i.e. 0, 2π, 4π,.....)
Furthur we take the most simple case of ¢ which is zero.
Thus the simplest eq for a particle having shm measured from mean position at start of time bcomes
x= A sin (wt)