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Chapter 10: Problem 34

Show that, for SHM, the maximum displacement, velocity, and acceleration are related by \(v_{\mathrm{m}}^{2}=a_{\mathrm{m}} A\).

Short Answer

Expert verified
Question: Prove the relationship \(v_{\mathrm{m}}^{2}=a_{\mathrm{m}} A\) for Simple Harmonic Motion by analyzing the maximum displacement (A), maximum velocity (v_m), and maximum acceleration (a_m). Answer: By analyzing the equations for displacement, velocity, and acceleration in Simple Harmonic Motion and finding their maximum values, we proved that the relationship \(v_{\mathrm{m}}^{2}=a_{\mathrm{m}} A\) holds true.

Step by step solution

01

Write down the equations for displacement, velocity, and acceleration in SHM

We'll start by writing down the equations for displacement (x), velocity (v), and acceleration (a) in SHM. Displacement: \(x(t) = A \cos(\omega t + \phi)\) Velocity: \(v(t) = -A\omega \sin(\omega t + \phi)\) Acceleration: \(a(t) = -A\omega^2 \cos(\omega t + \phi)\) Here, A represents the amplitude, \(\omega\) is the angular frequency, and \(\phi\) is the phase.
02

Calculate the maximum velocity

In order to find the maximum velocity (v_m), we need to find the maximum value of the velocity function \(v(t)\). This would be when the sine function is equal to its maximum value (±1). Therefore, \(v_{\mathrm{m}}=\pm A\omega\)
03

Calculate the maximum acceleration

Similar to Step 2, in order to find the maximum acceleration (a_m), we need to find the maximum value of the acceleration function \(a(t)\). This would be when the cosine function is equal to its maximum value (±1). Therefore, \(a_{\mathrm{m}}=\pm A\omega^{2}\)
04

Show that \(v_{\mathrm{m}}^{2}=a_{\mathrm{m}} A\)

Now, we will substitute the expressions for \(v_{\mathrm{m}}\) and \(a_{\mathrm{m}}\) we calculated in Steps 2 and 3 into the given equation. \(v_{\mathrm{m}}^{2} = (\pm A\omega)^{2} = a_{\mathrm{m}} A = (\pm A\omega^{2})A\) Simplify and compare: \((\pm A\omega)^{2} = (\pm A\omega^{2})A\) \((\pm A^2\omega^2) = (\pm A^2\omega^2)\) Hence, the given relationship \(v_{\mathrm{m}}^{2}=a_{\mathrm{m}} A\) is proved for Simple Harmonic Motion.

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