Problem 2-1D motion and potential energy A point particle with mass M is subject to a conservative force, F(x), directed along the x-axis. The potential energy associated with this force is U(x) = a/x². b/x, with a and b positive parameters. (a) Find the value x-xo corresponding to the equilibrium point and discuss whether this is a stable or unstable equilibrium. Sketch a graph of U(x). (b) Find the value, Emin, of the mechanical energy and show that Emin is negative. Find the minimum and maximum values of x reached by the particle for negative values of the mechanical energy such that Emin < E<0. (c) Assuming that the point particle stays all the time very close to xo, obtain an expression for the frequency of small oscillations around the equilibrium point.
Problem 2-1D motion and potential energy A point particle with mass M is subject to a conservative force, F(x), directed along the x-axis. The potential energy associated with this force is U(x) = a/x². b/x, with a and b positive parameters. (a) Find the value x-xo corresponding to the equilibrium point and discuss whether this is a stable or unstable equilibrium. Sketch a graph of U(x). (b) Find the value, Emin, of the mechanical energy and show that Emin is negative. Find the minimum and maximum values of x reached by the particle for negative values of the mechanical energy such that Emin < E<0. (c) Assuming that the point particle stays all the time very close to xo, obtain an expression for the frequency of small oscillations around the equilibrium point.
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In this question I don’t understand why they integrated U(x) what’s the purpose behind that and then part b if you can explain it in steps please because their explanation doesn’t make sense
Transcribed Image Text:= 1.3x10³ kg/s for the drag coefficient, calculate the power associated WILLI
Problem 2-1D motion and potential energy
A point particle with mass M is subject to a conservative force, F(x), directed along the x-axis. The potential energy associated with this force is U(x) = a/x² -
b/x, with a and b positive parameters. (a) Find the value x-xo corresponding to the equilibrium point and discuss whether this is a stable or unstable
equilibrium. Sketch a graph of U(x). (b) Find the value, Emin, of the mechanical energy and show that Emin is negative. Find the minimum and maximum values
of x reached by the particle for negative values of the mechanical energy such that Emin < E<0. (c) Assuming that the point particle stays all the time very
close to xo, obtain an expression for the frequency of small oscillations around the equilibrium point.
tation kinetic energy
the figuro). The nulley is a uniform cylinder with
Transcribed Image Text:Problem 2 - 10 motion and potential energy
(a) The equilibrium point,
2a
du
dx
x
0
Min
+
The value of U(x) et 26 is
U(₂) =
b
x²
ما
x=71
Umin
<0,
because a and b are positive. A sketch
for U(a) is as follows
U(₂) d
= 0 => x=
satisfies
b
2-2=-6²
26
Clearly, Simce хо
is a minimum of
U(₂), it is a
stable equilibrium
point.
MacBook Air
(b) The mechanical
energy is
E = K+U. the minimum value of E
is obtained for kinetic energy K=0
and
U = Umen. Thus:
Umin < 0.
*
Emin
When Emin <E<O, the minimum and
maximum values of x are reached when K=O
and therefore satisfy
-2-
E = _a_ b
x²
2
This is a quadratic polynomial for x,
whose solutions are
b
x = ±
2E
2
2
E
Since Eco, both I roots are positive,
+
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Step 1: Give data:
VIEWStep 2: a) Determination of x=x0 corresponding to the equilibrium point:
VIEWStep 3: Explanation of stable and unstable equilibrium:
VIEWStep 4: Sketch of U(x) vs x:
VIEWStep 5: (b) Determination of minimum value of mechanical energy:
VIEWStep 6: Calculation of minimum and maximum values of x for E(min)<E<0:
VIEWStep 7: Calculation of frequency of small oscillation about equilibrium position:
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