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Question

# The potential energy of a particle is given by relation U=x3−6x2 (In SI units). The nature of equilibrium at x=0 is

A
Stable
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B
Unstable
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C
Neutral
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D
Undefined
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Solution

## The correct option is B Unstable It is given that, U=x3−6x2 We also know that F=−dUdx=−3x2+12x ∴dUdx=3x2−12x For the condition of equilibrium, force F=0⇒F=−3x2+12x=0 ⇒F=0⇒3x(4−x)=0 ∴x=0 ,x=4 d2Udx2=ddx(3x2−12x)=6x−12 At (x=0) ∴d2Udx2=−12<0 ∵F=0 & d2Udx2<0 is the condition for unstable equilibrium ⇒ Unstable Equillibrium

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