Question
S.H.M. displacement of a particle in $x=A$ $\sin (\omega t+\phi)$ is given by. So that the value of $x$ remains same if the time increased by $\frac{2 \pi}{\omega}$.
✓
Answer
S.H.M. displacement of particle in at time
$ x=A \sin (\omega t+\phi) $
If the time $\left( T +\frac{2 \pi}{\omega}\right)$ then the particle displacement is $x_1$, then
$ \begin{array}{l} x_1=A \sin \left(\omega\left(t+\frac{2 \pi}{\omega}\right)+\phi\right) \\ x_1=A \sin [(\omega t+2 \pi)+\phi]=A \sin [2 \pi+(\omega t+\phi)] \\ x_1=A \sin (\omega t+\phi)=x \\ x_1=x \end{array} $
So, the time in S.H.M. is remains same.
$ x=A \sin (\omega t+\phi) $
If the time $\left( T +\frac{2 \pi}{\omega}\right)$ then the particle displacement is $x_1$, then
$ \begin{array}{l} x_1=A \sin \left(\omega\left(t+\frac{2 \pi}{\omega}\right)+\phi\right) \\ x_1=A \sin [(\omega t+2 \pi)+\phi]=A \sin [2 \pi+(\omega t+\phi)] \\ x_1=A \sin (\omega t+\phi)=x \\ x_1=x \end{array} $
So, the time in S.H.M. is remains same.
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