Question

Apply the Binet equation to solve the motion of a particle in a force field F = -kr. In which case is an harmonic solution obtained?

Answer 1

Answer:

$\vec r = r_o sin(\omega t + \phi)$

here we know that

$\omega = \sqrt{\frac{k}{m}}$

And this equation is the harmonic solution of the motion

Explanation:

As we know by Newton's II law

$F_{net} = ma$

also we know that

$\vec F = -k\vec r$

so we will have

$m\frac{d^2r}{dt^2} = - kr$

so we will have

$\frac{d^2r}{dt^2} + \frac{k}{m} r = 0$

now we know that

$\frac{k}{m} = constant = \omega^2$

so we have

$\frac{d^2r}{dt^2} + \omega^2 r = 0$

so the solution of this equation would be

$\vec r = r_o sin(\omega t + \phi)$

here we know that

$\omega = \sqrt{\frac{k}{m}}$

And this equation is the harmonic solution of the motion

#Learn

Topic : Harmonic Motion

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