The total energy of a particle is given by:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]E_{tot} = 2 \dot{x}^{2} - cos(\frac{1}{2} \pi x)[/tex]

and i'm told that the particle passed through the point [tex]x=1 [m][/tex] with a velocity of [tex]\vec{v}=-\frac{1}{2} \hat{x} [m/sec][/tex].

I'm required to find the maximum velocity of the particle during its motion.

The attempt at a solution

well I know that at equilibrium, the total energy of the particle is kinetic energy, so this corresponds to the maximum velocity.

Equilibrium occurs when the force on the body, i.e. the first derivative of the potential energy term with respect to x, vanishes.

The P.E. is extracted from the total energy expression, that is,

[tex] U(x)= - cos(\frac{1}{2} \pi x) [/tex]

so

U'(x) = 0 ===> equilibrium points are: [tex]x_{e} = 2n , n=0,1,2,... [/tex]

and at this point i'm stuck!

Is it even the right approach to solve questions like these, or not?!

by the way, the answer is:

[tex] v_{max} = \sqrt{\frac{3}{4}} [m/sec] [/tex]

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# Homework Help: Energy & initial conditions

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