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Question

The potential energy of a particle in a force field is given by the equation E=x334x2+15x6 Choose the correct statement among the options.

A
no points of equilibrium exist.
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B
x=3 and x=5 are points of equilibrium.
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C
There is a stable equilibrium at x=3 and unstable equilibrium at x=5.
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D
There is neutral equilibrium at x=3 and x=5.
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Solution

The correct option is B x=3 and x=5 are points of equilibrium.
Given,
E=x334x2+15x6
For equilibrium,
dEdx=0
Therefore, we get,
dEdx=ddx(x334x2+15x6)=03x238x+15=0x28x+15=0(x3)(x5)=0x=3 or x=5

Hence the body will be in equilibrium at x=3 and x=5.

If d2Edx2>0, then it is a stable equilibrium, If d2Edx2<0, then it is an unstable equilibrium.
d2Edx2=2x8
At x=3, d2Edx2=2×38=2

The system is in an unstable equilibrium at x=3

At x=5,
d2Edx2=2×58=2
The system is in a stable equilibrium at x=5.

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