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Question

Potential energy function along positive x-axis is given by
U(x)=ax+bx (only for x>0) where a and b are constants. It is known that system has only one stable equilibrium configuration. Find the possible values of a and b. Justify your answer.

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Solution

U(x)=ax+bx
F=dUdt=(abx2)=a+bx2
F=0 for equilibrium,a+bx2=0
For stable equilibrium ,dUdt2<0
2bx3<0 here x>0 then,
b>0,a=bx2

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