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If an object suspended from a spring is displaced vertically from its equilibrium position by a small amount and re- leased, and if the air resistance and the mass of the spring are ignored, then the resulting oscillation of the object is called simple harmonic motion. Under appropriate condi- tions the displacement y from equilibrium in terms of time t is given by $$y=A\cos \omega t$$ where A is the initial displacement at time t = 0, and ω is a constant that depends on the mass of the object and the stiffness of the spring (see the accompanying figure). The constant |A| is called the amplitude of the motion and ω the angular frequency..... The period T is the time required to make one complete oscillation. Show that T = 2π/ω.

If I take amplitude equal to $2\pi$ than, I found an equation which is nearly related to period. But, I noticed there's a $\cos$. I don't know how to remove it. How to solve it? I would request for hint.

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  • $\begingroup$ How are you taking the amplitude equal to $2\pi$? Could you please elaborate about it in the question? $\endgroup$
    – Deep Bhowmik
    May 29, 2021 at 16:07

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Time period is defined as the time interval taken for the system to finish one complete oscillation. This means that the phase difference between the initial and final position is $2\pi$. This means that: $$\omega \Delta t=2\pi$$ so that $$\Delta t=\frac {2\pi}{\omega}$$ Remember that, amplitude is not a property of a system, it depends upon the external forces acting upon the system. Hence you cannot simply define it to be $2\pi$.

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  • $\begingroup$ But, what about $\cos$? How we can remove it? Where does it go? $\endgroup$
    – user876873
    May 29, 2021 at 16:24
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    $\begingroup$ It doesn't "go" anywhere. Just that the phase changes by $2\pi$, since $\cos (\theta)=\cos(\theta+ 2\pi)$ $\endgroup$
    – Ritam_Dasgupta
    May 29, 2021 at 16:27

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