(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A particle P F=36/x^3-9/x^2 (X>0)

Show that each motion of P consists of either(i) a periodic oscillation between two extreeme points , or (ii) an unbounded motion with one extreme point, depending upon the value of the total energy . Initially P is projected from the point x=4 with speed 0.5

. Show that P oscillates between two extremes points and find the period of the motion.

[You may make use of the formula

Show that there is a single equilibrium position for P and that it is stable. Find the period of small oscillations about this point.

2. Relevant equations

.5*m*v^2+V(x)=E

-dV/dx=F(x)

3. The attempt at a solution

V(x)=-F*dx=

V(x)=18/x^2-9/x

.5*v^2+18/x^2-9/x=E(x)

plugging v=.5 and x=4.0m

.5*(.5)^2+18/(4)^2-9/4=E

tau=2*[tex]\int[/tex] dx/[2*(E-V(x))]^{.5}

E=-1.00

therefore

v^2=-1-(18/x^2)+9/x

plugging in v=.5 and x=4 I can now find the extreme points ;

for the particle to be stable d^2V/dx^2 >0

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# Unconstrained motion

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