The diagram shows a plot of the potential energy as a function of x for a particle moving along the x axis. The points of stable equilibrium are: d e only c (A) only a B) only b (C) (D) only d
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Answer:
Solution
To find the allowed regions for x, we use the condition
K
=
E
−
U
=
−
1
4
−
2
(
x
4
−
x
2
)
≥
0.
If we complete the square in
x
2
, this condition simplifies to
2
(
x
2
−
1
2
)
2
≤
1
4
,
which we can solve to obtain
1
2
−
√
1
8
≤
x
2
≤
1
2
+
√
1
8
.
This represents two allowed regions,
x
p
≤
x
≤
x
R
and
−
x
R
≤
x
≤
−
x
p
,
where
x
p
=
0.38
and
x
R
=
0.92
(in meters).
To find the equilibrium points, we solve the equation
d
U
/
d
x
=
8
x
3
−
4
x
=
0
and find
x
=
0
and
x
=
±
x
Q
, where
x
Q
=
1
/
√
2
=
0.707
(meters). The second derivative
d
2
U
/
d
x
2
=
24
x
2
−
4
is negative at
x
=
0
, so that position is a relative maximum and the equilibrium there is unstable. The second derivative is positive at
x
=
±
x
Q
, so these positions are relative minima and represent stable equilibria
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