Question 130: Work, Energy and Power & Circular Motion

Question

If the potential energy between two molecules is given by
U = $ - {A \over {{r^6}}} + {B \over {{r^{12}}}}$,
then at equilibrium,
separation between molecules, and the potential energy are :

Accepted Answer

U = $ - {A \over {{r^6}}} + {B \over {{r^{12}}}}$
F = - ${{dU} \over {dr}}$
= Ã¢ÂÂ (A(Ã¢ÂÂ6r
Ã¢ÂÂ7
)) + B(Ã¢ÂÂ12r
Ã¢ÂÂ13
)
for equilibrium, F = 0
$ 	herefore $ 0 = ${{6A} \over {{r^7}}} - {{12B} \over {{r^{13}}}}$
$ \Rightarrow $ ${{6A} \over {12B}} = {1 \over {{r^6}}}$
$ \Rightarrow $ r = ${\left( {{{2B} \over A}} ight)^{{1 \over 6}}}$
$ 	herefore $ U = $ - {A \over {{{2B} \over A}}} + {B \over {{{\left( {{{2B} \over A}} ight)}^2}}}$
= $ - {{{A^2}} \over {2B}} + {{{A^2}} \over {4B}}$
= $ - {{{A^2}} \over {4B}}$

Answer

${\left( {{{2B} \over A}} \right)^{1/6}}$, $ - {{{A^2}} \over {4B}}$

Answer

${\left( {{{2B} \over A}} \right)^{1/6}}, - {{{A^2}} \over {2B}}$

Answer

${\left( {{B \over A}} \right)^{1/6}},0$

Answer

${\left( {{B \over {2A}}} \right)^{1/6}}, - {{{A^2}} \over {2B}}$

Work, Energy and Power & Circular Motion

If the potential energy between two molecules is given by U = - {A \over {{r^6}}} + {B \over {{r^{12}}}}, then at equilibrium, separation between molecules, and the potential energy are :

Choose the correct answer: