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Question

Potential energy of a system of particles is U=α3r3β2r2, where r is distance between the particles. Here α and β are positive constants. Which of the following are correct for the system?


Correct Answer
A
Equilibrium separation between the particles is αβ
Correct Answer
B
For r=αβ, the equilibrium is stable
Your Answer
C
For r=αβ, the equilibrium is unstable
Your Answer
D
Work required to slowly move the particles to infinite separation from initial equilibrium position is βα

Solution

The correct options are
A Equilibrium separation between the particles is αβ
B For r=αβ, the equilibrium is stable
Given: U=α3r3β2r2
For equilibrium condition, F=0
dUdr=0
[3α3r42β2r3]=0
αr4βr3=0
req=αβ

Now,
d2Udr2=+4αr53βr4
At r=αβ,
d2Udr2=4αβ5α53ββ4α4=β5α4
Since d2Udr2>0
stable equilibrium (as P.E is minimum)

Work required to slowly move the particle from r=αβ to
=UUr=αβ
=0β36α2
=β36α2

Option A and B are correct.

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