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5.work,energy,power and collision — Physics STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 SciencePhysics5.work,energy,power and collision1 Mark
MCQ
The potential energy between two atoms in a molecule is given by $U(x) = \frac{a}{{{x^{12}}}} - \frac{b}{{{x^6}}}$; where $a$ and $b$ are positive constants and $x$  is the distance between the atoms. The atom is in stable equilibrium when
  • A
    $x = \sqrt[6]{{\frac{{11a}}{{5b}}}}$
  • B
    $x = \sqrt[6]{{\frac{a}{{2b}}}}$
  • C
    $x = 0$
  • ✓
    $x = \sqrt[6]{{\frac{{2a}}{b}}}$
✓

Answer

Correct option: D.
$x = \sqrt[6]{{\frac{{2a}}{b}}}$
d
(d)Condition for stable equilibrium $F = - \frac{{dU}}{{dx}} = 0$ 

$⇒$ $ - \frac{d}{{dx}}\left[ {\frac{a}{{{x^{12}}}} - \frac{b}{{{x^6}}}} \right] = 0$

$⇒$ $ - 12a{x^{ - 13}} + 6b{x^{ - 7}} = 0$

$⇒$ $\frac{{12a}}{{{x^{13}}}} = \frac{{6b}}{{{x^7}}}$

$⇒$ $\frac{{2a}}{b} = {x^6}$

$⇒$  $x = \sqrt[6]{{\frac{{2a}}{b}}}$

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