GREPhysics.NET: GR8677 #60

GR8677 #60

Problem

GREPhysics.NET Official Solution

Mechanics $\Rightarrow$ }Simple Harmonic Motion

Recall $F=-\frac{\partial V}{\partial x}=-2bx$ . Simple harmonic motion has the simple form, $\ddot{x}\propto x$ . Thus $F=m\ddot{x}=-2bx \Rightarrow \ddot{x}=-\omega^2 x$ , where $\omega^2=2b/m$ is the frequency squared. Thus, simple harmonic motion occurs with a frequency determined by both $b$ and $m$ . This is choice (C).

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Comments

ernest21
2019-08-23 02:03:37

I enjoyed reading your Solutions together with the answers of others. This is truly a great read for me. brick breaker 2

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a62
2016-09-20 02:36:59

Easy, but even easier if you look at it this way: a is absolutely meaningless because the zero of potential is arbitrary. So it\'s exactly like the mass-on-a-spring model.

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Almno10
2010-11-11 23:02:29

Force is the negative gradient of potential energy. The a drops out.

psychonautQQ
2013-09-26 08:41:23

but this is asking for the frequency, how does force and frequency relate?

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sirius
2008-11-05 22:26:45

A similar solution if you remember $V(x) = \frac{1}{2}m\omega^2x^2$ . The given V(x) has a vertical shift, a, which can be ignored by shifting your zero-point energy. So, $b=\frac{1}{2}m\omega^2$ , solving for $\omega$ makes it depend on b and m.

This is enough, but the problem asks for frequency $f=\frac{\omega}{2\pi}$ . So $f=\frac{1}{2\pi}\sqrt{\frac{b}{2m}}$ . making f depend on b and m. The answer then is (C).

alemsalem
2010-09-21 06:25:37

i totally agree, to be sure that it doesn't depend on a, just remember that shifting the potential energy by a constant cannot change the motion (classically) so it doesn't affect the frequency.

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A similar solution if you remember $V(x) = \frac{1}{2}m\omega^2x^2$ . The given V(x) has a vertical shift, a, which can be ignored by shifting your zero-point energy. So, $b=\frac{1}{2}m\omega^2$ , solving for $\omega$ makes it depend on b and m. This is enough, but the problem asks for frequency $f=\frac{\omega}{2\pi}$ . So $f=\frac{1}{2\pi}\sqrt{\frac{b}{2m}}$ . making f depend on b and m. The answer then is (C).

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