Physics 1116 Fall 2011 Homework 2 Due: Wednesday, Sep 7

Reading: K&K §1.9 and Note 1.1; K&K §§2.1-2.4(The problems in this set focus on K&K §1.9 and Note 1.1. §§2.1-2.4 is mainly an introductionto Newton’s laws that you should read and think about.)

1. Some Mathematical Tools

Read Note 1.1 in K&K and (if you need to) review Taylor series. Then do the followingexercises.

a. Write down the Taylor series for each of the following functions, and give thesimplest nontrivial approximation valid for small x.

(i) 3√

1 + x− 5√

1 + x

(ii) ln(1 + x)

(iii) e−x2

(iv) sinhx

(v) cosh x

Note: sinhx ≡ (ex − e−x)/2; cosh x ≡ (ex + e−x)/2.

b. It’s handy to have a rough idea where common approximations breakdown. Writedown simple algebraic expressions for the errors incurred in making the approxima-tions sinx ≈ x and cosx ≈ 1− 1

2x2. In each case give a numerical value in degrees

for the angle at which this error reaches 10% of the correct, unapproximated, value.

c. In Special Relativity, as we will see, the total energy E of a particle of rest mass mand momentum p is given by E =

√p2c2 +m2c4. The realm of Newtonian mechan-

ics is where pc� mc2. In this domain, find the simplest nontrivial approximationfor E, consisting of two terms, and explain what the two terms mean.

2. Approximate Harmonic Potentials.

Harmonic potentials are quadratic functions of the form U(x) = 12kx2 which play an

important role in oscillatory systems. You might recognize this function as the potentialenergy for a spring, of spring constant k, stretched a distance x beyond its restingposition. (But if you don’t, don’t worry.) In this problem we will find that harmonicpotentials turn up in many places, once you know how to look.

a. The circular hoop.

Picture a bead resting at the bottom of a frictionless circular hoop of radius r.The plane of the hoop is vertical. For definiteness, let the hoop be defined by theequation for the circle x2 +(y−r)2 = r2, with y measuring vertical height above theminimum point, and x measuring horizontal position. Since gravitational potentialenergy is given by U(y) = mgy for a bead of mass m, show that for small enoughexcursions around the bottom of the circle, U(x) ≈ 1

2kx2. Find an expression for k

in terms of the available parameters.

b. The Lennard-Jones inter-molecular potential.

The interaction between two molecules that attract one another and form a boundstate, may be approximated by a potential function

U(r) =a

r12− b

r6.

(i) If U has units of energy, what are the units of a and b?

(ii) Plot U(r) and observe the shape. (For this purpose, set a = 1, b = 2 in theirrespective units. )

(iii) Compute the location, r0, of the minimum of U(r) in terms of arbitrary a andb.

(iv) Calculate the Taylor Series expansion of U(r) about the minimum point, andtruncate the series under the assumption that you’re only interested in theregion where |r − r0| is small. This should give something of the form U(r) =U0 + 1

2k(r − r0)2 ; identify the variable k in terms of the given parameters.

c. The Higgs potential.

Consider a one-dimensional toy “Higgs potential”, U(x) = −12µ2x2 + 1

4λ2x4.

(i) Plot U(x) for −2 < x < 2 and observe the shape. (For this purpose, setµ2 = λ2 = 1.)

(ii) Compute the location, x0 > 0, of the positive-side minimum of U(x) in termsof arbitrary µ and λ.

(iii) Calculate the Taylor Series expansion of U(x) about the minimum point, andtruncate the series under the assumption that you’re only interested in theregion where |x− x0| is small. This should give something of the form U(x) =U0 + 1

2k(x− x0)

2 ; identify the variable k in terms of the given parameters.

3. Complex exponentials and circular motion.

a. Write down the Taylor series for eix (i =√−1), and then rewrite this as the sum

of two other Taylor series, the first one composed of all the even-power terms (x2,x4, etc) and the second composed of all the odd-power terms. Observe that thisexpression has a very simple equivalent in familiar functions.

b. A complex number like z = a+ ib is conveniently represented in the complex planein much the same as a vector r = ax + by is represented in the real xy plane. Theanalog of r2 = r · r is |z|2 = zz∗ = (a + ib)(a − ib). In the complex plane, thehorizontal axis measures the real part of a number and the vertical axis measuresthe imaginary part; in the real plane the horizontal and vertical axes measure the“x” and “y” components of a vector. In both cases we can also use plane polarcoordinates, (r, θ). Show by explicit construction that any complex number a + ibcan be written as reiθ. In other words, compute r and θ in terms of a and b.

c. Since an object in uniform circular motion can be represented in xy Cartesiancoordinates by r = r(cosωtx+sinωty), we can see that it might also be convenientlyrepresented by r(t) = r0e

iωt in the complex plane if we let the imaginary axis

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play the role of the y axis of the real plane. From this, derive the velocity andacceleration and show that the results agree with standard statements about UCM.

4. Spiral motion.

K.K. problem 1.20

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