Physics 227 Homework II - Autumn 2008, Assignments of Physics

Problem solutions for physics 227, a university-level physics course, from autumn 2008. The homework includes problems on potential energy, oscillations, and electric potential. Students are required to check their results using mathematica.

Typology: Assignments

Pre 2010

Uploaded on 03/18/2009

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Physics 227 HW II 1 Autumn 2008
Physics 227 - Autumn 2008
HW II
Due Friday 10/10/08
Except as noted, all problems are in the class text, Boas.
Example problems: Not to be turned in solutions can be found at the back of the
book or in the companion volume of solved problems. See also Appendix B to the
relevant lecture.
§1.13: (11 in Ed. 2), 27 (15 in Ed. 2)
§1.15: 17
§2.4: 11, 17
§2.5: 6, 43
Assigned problems: To be turned in. Problems A and B below will be graded like
the exercises from Boas. All graded problems are worth 5 points each. Problem C is
extra credit: doing extra credit problems do not change your score, but at the end of
the term can make the difference in borderline grade assignments.
§1.13: 26 (12 in Ed. 2) [Note that you are asked to check your result using
Mathematica. Please include enough output from your computer work to confirm
that you checked your work. Note also that Mathematica has a useful command
Series[].]
§1.15: 23(d) (21 in Ed. 2) [Again check your result using Mathematica.], 28
§2.4: 4 [Again check your result using Mathematica.]
§2.5: 33 [Again check your result using Mathematica.]
A. (5 points) Remember that a particle of mass M on a spring with spring constant k
and equilibrium length
0
x
obeys Newton's law
,
dU
Mx dx


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Physics 227 - Autumn 2008

HW II

Due Friday 10/10/

Except as noted, all problems are in the class text, Boas.

Example problems: Not to be turned in – solutions can be found at the back of the book or in the companion volume of solved problems. See also Appendix B to the relevant lecture.

§1.13: (11 in Ed. 2), 27 (15 in Ed. 2) §1.15: 17 §2.4: 11, 17 §2.5: 6, 43

Assigned problems: To be turned in. Problems A and B below will be graded like the exercises from Boas. All graded problems are worth 5 points each. Problem C is extra credit: doing extra credit problems do not change your score, but at the end of the term can make the difference in borderline grade assignments.

§1.13: 26 (12 in Ed. 2) [Note that you are asked to check your result using Mathematica. Please include enough output from your computer work to confirm that you checked your work. Note also that Mathematica has a useful command Series[].] §1.15: 23(d) (21 in Ed. 2) [Again check your result using Mathematica .], 28 §2.4: 4 [Again check your result using Mathematica .] §2.5: 33 [Again check your result using Mathematica .]

A. (5 points) Remember that a particle of mass M on a spring with spring constant k and equilibrium length x 0 obeys Newton's law

dU

Mx

dx

where U  x is the potential energy,

2 0 0

U x  U  k x  x

and where the value of the constant (^) U 0 is irrelevant to the particle's motion. You have seen previously that the resultant motion is a steady oscillation with frequency

  k M.

Suppose instead that we consider the motion of a particle of mass M in a potential

a b

U x

x x

where a and b are positive dimensionful constants.

a. Sketch the potential U  x for x  0 and check with a Mathematic a plot for a

= b = 1

b. Find the “equilibrium position” x 0 in terms of a and b. This is the position where the particle feels no force. Check your result using Mathematica.

c. Perform a Taylor expansion of U  x about x  x 0 to order  

2 xx 0 and check your expansion with Mathematica. Compare your approximate expression for U with the example of a particle on a spring, and thereby determine the oscillation frequency of the particle if it makes small

oscillations in the potential U  x about the equilibrium position x 0

B. (5 points) Suppose four charges are placed on the x -axis: charges + Q are located at positions x = +2 a and x = -2 a , while charges – Q are located at positions x = + a and x = - a.

  • Q - Q - Q + Q

x = -2 a - a 0 a 2 a X