A particle of mass m moves along the x-axis such that its acceleration is d²x dt² = 8x³. - -X. (a) Determine the potential energy, U(x), if U = 0 at x = 0. Define the total energy E of the particle. Calculate dE/dt and conclude that E is a constant of motion. (b) Show that the system has two unstable equilibrium points at x = ±₁₂1 and a stable equilibrium point at x = 0. Sketch the graph of U(x). 2√2 (c) The particle is initially at the origin with velocity vo. Show that the particle can escape to only if vol> //.

A particle of mass m moves along the x-axis such that its acceleration is d²x dt² = 8x³. - -X. (a) Determine the potential energy, U(x), if U = 0 at x = 0. Define the total energy E of the particle. Calculate dE/dt and conclude that E is a constant of motion. (b) Show that the system has two unstable equilibrium points at x = ±₁₂1 and a stable equilibrium point at x = 0. Sketch the graph of U(x). 2√2 (c) The particle is initially at the origin with velocity vo. Show that the particle can escape to only if vol> //.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

A particle of mass m moves along the x-axis such that its acceleration is
d²x
dt²
=
8x³ - x.
(a) Determine the potential energy, U(x), if U = 0 at x = 0. Define the total energy E of
the particle. Calculate dE/dt and conclude that E is a constant of motion.
(b) Show that the system has two unstable equilibrium points at x =
equilibrium point at x = 0. Sketch the graph of U (x).
1
2√2
=+ and a stable
(c) The particle is initially at the origin with velocity vo. Show that the particle can escape
to ±~ only if |vo| > 4

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