Chapter 10: Q. 57 (page 259)

A system has potential energy
$U (x) = x + \sin ((2 r a d / m) x)$
as a particle moves over the range $0 m \leq x \leq π m$ .
a. Where are the equilibrium positions in this range?
b. For each, is it a point of stable or unstable equilibrium?

Short Answer

Expert verified

(a) The equilibrium position in the range $0 m \leq x \leq π m x = \frac{π}{3}, \frac{2 π}{3}$ .

(b) The equilibrium position $x = \frac{π}{3}$ is unstable and $x = \frac{2 π}{3}$ is a stable equilibrium.

Step by step solution

Given information (part a)

A system has potential energy $U (x) = x + \sin (2 x (rad))$ , where x is in m,as the particle moves over the range $0 m \leq x \leq π m$ .

Explanation (part a)

To find the equilibrium positions over the range $0 m \leq x \leq π m$ ,

$U (x) = x + \sin (2 x) - \frac{d U (x)}{d x} = 0 - \frac{d [x + \sin (2 x)]}{d x} = 0 1 + 2 \cos (2 x) = 0 \cos (2 x) = - \frac{1}{2} 2 x = \frac{2 π}{3}, \frac{4 π}{3} x = \frac{π}{3}, \frac{2 π}{3}$

Given information (part b)

A system has potential energy $U (x) = x + \sin (2 x (rad))$ , wherexis inm, as the particle moves over the range $0 m \leq x \leq π m$ .

Explanation (part b)

To determine the stability,

$- \frac{d^{2} U}{d x^{2}} = - \frac{d^{2} [x + \sin (2 x)]}{d x^{2}} - \frac{d^{2} U}{d x^{2}} = - \frac{d [1 + 2 \cos (2 x)]}{d x} - \frac{d^{2} U}{d x^{2}} = - [- 4 \sin (2 x)] - \frac{d^{2} U}{d x^{2}} = 4 \sin (2 x) - {\frac{d^{2} U}{d x^{2}}|}_{x = \frac{π}{3}} = 4 \sin (\frac{2 π}{3}) - {\frac{d^{2} U}{d x^{2}}|}_{x = \frac{π}{3}} = 2 \sqrt{3} > 0$

And,

$- \frac{d^{2} U}{d x^{2}} = 4 \sin (2 x) - {\frac{d^{2} U}{d x^{2}}|}_{x = \frac{π}{3}} = 4 \sin (\frac{4 π}{3}) - {\frac{d^{2} U}{d x^{2}}|}_{x = \frac{π}{2}} = - 2 \sqrt{3} < 0$

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A system has potential energy
$U (x) = x + \sin ((2 r a d / m) x)$
as a particle moves over the range $0 m \leq x \leq π m$ .
a. Where are the equilibrium positions in this range?
b. For each, is it a point of stable or unstable equilibrium?

Short Answer

Step by step solution

Given information (part a)

Explanation (part a)

Given information (part b)

Explanation (part b)

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A system has potential energyUx=x+sin2rad/mxas a particle moves over the range0m≤x≤πm.a. Where are the equilibrium positions in this range?b. For each, is it a point of stable or unstable equilibrium?

Short Answer

Step by step solution

Given information (part a)

Explanation (part a)

Given information (part b)

Explanation (part b)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism

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Study anywhere. Anytime. Across all devices.

A system has potential energy
$U (x) = x + \sin ((2 r a d / m) x)$
as a particle moves over the range $0 m \leq x \leq π m$ .
a. Where are the equilibrium positions in this range?
b. For each, is it a point of stable or unstable equilibrium?