Question
The maximum acceleration of a simple harmonic oscillator is $\text{a}_0$ and the maximum velocity is $\text{v}_0$. What is the displacement amplitude?
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Answer
Let A be the displacement amplitude and $\omega$ be the angular frequency of the simple harmonic oscillator. Then, $\text{a}_0=\omega^2\text{A}$$\text{v}_0=\omega\text{A}$
On dividing $\frac{\text{v}_0^2}{\text{a}_0}=\frac{\omega^2\text{A}^2}{\omega^2\text{A}}=\text{A}$$\text{A}=\frac{\text{v}_0^2}{\text{a}_0}$
On dividing $\frac{\text{v}_0^2}{\text{a}_0}=\frac{\omega^2\text{A}^2}{\omega^2\text{A}}=\text{A}$$\text{A}=\frac{\text{v}_0^2}{\text{a}_0}$
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