(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] Approximation to simple harmonic motion.

1. The problem statement, all variables and given/known data

A small mass [tex]m[/tex], which carries a charge [tex]q[/tex], is constrained to move vertically inside a narrow, frictionless cylinder. At the bottom of the cylinder is a point mass of charge [tex]Q[/tex] having the same sign as [tex]q[/tex]. Show that if the mass [tex]m[/tex] is displaced by a small amount from its equilibrium position and released, it will exhibit simple harmonic motion with angular frequency

[tex]\omega = \sqrt{\frac{2g}{y_0}}[/tex]

where [tex]y_0[/tex] is the height, when the mass is in equilibrium.

2. Relevant equations

Simple harmonic motion:

[tex]F=-kx[/tex]

Coulomb's Law(one dimension, same sign of charges):

[tex]F= \frac{kq_1q_2}{y^2}[/tex]

Newton's Second Law:

[tex]F=ma[/tex]

3. The attempt at a solution

Force acting on the mass [tex]m[/tex] at a height [tex]y[/tex] is:

[tex]F=\frac{kqQ}{y^2} - mg[/tex]

I found the potential energy by:

[tex] U=-\int F dy= \frac{kqQ}{y} + mgy [/tex]

I found the equilibrium point:

[tex] mg = \frac{kqQ}{y_0^2} [/tex]

[tex] y_0 = \sqrt{\frac{kqQ}{mg}} [/tex]

Near (stable) equilibrium I can approximate the potential energy by some parabola with equation:

[tex]U_n = U(y_0) + B(y - y_0)^2 = 2\sqrt{kqQmg} + B(y - \sqrt{\frac{kQq}{mg}})^2[/tex]

where [tex]B[/tex] is some (positive) constant.

Than I can find the approximate force by taking the negative derivative of [tex]U_n[/tex]. That is

[tex]F = -2B(y - y_0)[/tex]

That is the same as simple harmonic motion and so we can say that

[tex]\omega = \sqrt{\frac{2B}{m}}[/tex]

Well my problem is that I have no idea how to find the [tex]B[/tex] or how to show that the [tex]B = \frac{mg}{y_0}[/tex]. The thing is I don't know how to mathematically describe that I want the parabola of the approximate potential energy function to be the most similar to the original function near equilibrium.

I will really appreciate any help from you.

Thanks.

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# Homework Help: Approximation to simple harmonic motion.

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