Consider a simple harmonic motion of time period T. Calculate the time taken for the displacement to change value from half the amplitude to the amplitude.

Explanation

Let particle is at equilibrium position at `t=0`
`x=Asin(omegat+phi)`
At `t=0, x=0` i.e., `sin(phi)=0impliesphi=0`
`x=Asinomegat`

Time taken by particle to move from O to P i.e., `x=A//2`
`x=AsinomegatimpliesA/2=Asinomegat`
`sinomegat=1/2=sinpi//6`
`omegat=pi//6implies(2pi)/(T)t=pi/6impliest=(T)/(12)`
Time taken by particle to move from O to A `=T//4`
Time from P to A i.e., `x=A//2` to `x=A`
`=T//4-T//12=T//6`