Question 120: 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: