The potential energy of a particle has the following radial dependence, U = (1 -

Question

The potential energy of a particle has the following radial dependence,

U = (1 - r/6) re^-(r/3) ....

a) Determine if there are any finite equilibrium points for this particle. If
there are finite equilibrium points, determine their locations.

I know you have to do a derivative here possibly, but this is the part i don't know how to do. please show work, it would be very helpful.

b) If you did find any finite equilibrium points for the particle in (a),
determine whether they are points of stable or unstable equilibrium

stable is where the graph is U and unstable is the top of where the graph is curved. i believe this is the second derivative and this i also need help computing.

Thank you very much!!!

Explanation / Answer

yes you are right with both the parts

a) U = (1-r/6)re^(-r/3) = (r - r2/6)e-r/3

dU/dr = (1-r/3)*e-r/3 - 1/3(r - r2/6)e-r/3

For equililbrium dU/dr = 0

=>1 - r/3 - r + r2/6 = 0

multiply with 6

=> r2 - 8r + 6 = 0

=>r2-4r - 2r + 6 = 0

=>(r-4)(r-2) = 0

r = 4 or r = 2

b)d2U/dr2= d(dU/dr)/dr = -1/3e-r/3 -1/3((1-r/3)*e-r/3 - 1/3(r - r2/6)e-r/3)

= -1/3e-r/3 - 1/3e-r/3 + r/9e-r/3 - 1/9*r*e-r/3 + r2/54*e-r/3

put value of r = 4 and r = 2 in above equation.

if the value is positive then its minima hence stable equilibrium

if the value is negative then it maxima hence unstable

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