PHY226

Phil Lightfoot 2008/9 Tutorial Questions - Page 1 of 10

PHY226 Tutorial Questions Week 2: Binomial Series and Complex numbers 1. (a) Find the first few terms in the binomial series expansion of (1 – x)-1/2 .

(b) In the theory of special relativity, an object moving with velocity v has mass 221 cv

mm o

−= where

m0 is the mass of the object at rest and c is the speed of light. The kinetic energy of the object is the difference between its total energy and its energy at rest: K = mc2 - m0c2. Using the series found in (a), show that when v << c, the relativistic expression for K agrees with the classical expression K = ½ mv2.

PHY226

Phil Lightfoot 2008/9 Tutorial Questions - Page 2 of 10

(c) What is 31

z if 364πi

ez = ?

Week 3: First and second order ODEs 2. Make sure you can state (hopefully instantly!) the general solution of the following 1st order ODEs:

(a) dx xdt

α= − , (b) dx xdt

β= , (c) dx i xdt

γ= , where α, β, γ are real positive constants.

In what physical situations might equations of forms (a) and (b) occur?

3. What is the solution of the equation 0)(=

dttdN subject to the condition that N = N0 when t = 0?

4. 2nd order ODEs. In lecture 4 we discussed the equations and solutions for the linear harmonic

oscillator and unstable equilibrium. Now consider the 1D time independent Schrodinger equation for the spatial wavefunction u(x) of a

particle of energy E in a constant potential V: uEVudx

udm

=+− 2

22

2h .

Which form does this equation have, and what do its solutions look like, (i) for V < E? (ii) for V > E?

PHY226

Phil Lightfoot 2008/9 Tutorial Questions - Page 3 of 10

PHY226

Phil Lightfoot 2008/9 Tutorial Questions - Page 4 of 10

Week 4: ODEs cont. 5. (a) Express the complex function Z(ω) = - ω2 + 2iγω + ω0

2 in the form Z(ω) = |Z(ω)|eiφ by finding expressions for |Z(ω)| and tanφ.

(b) In the lecture we said that if X(t) is a solution of the equation cXdtdXb

dtXda ++2

2

= Feiωt then

x(t) = Re[X(t)] is the solution of tFcxdtdxb

dtxda ωcos2

2

=++ . Can you justify this?

PHY226

Phil Lightfoot 2008/9 Tutorial Questions - Page 5 of 10

(c) Using complex exponentials, find the steady state solution of the damped, driven oscillator

equation tFxdtdx

dtxd ωωγ cos2 2

02

2

=++ .

PHY226

Phil Lightfoot 2008/9 Tutorial Questions - Page 6 of 10

Week 5: Fourier Series 6. Show that the function shown has Fourier series

∑=oddn

nxn

xf sin14)(π

Explain why the series only contains sine terms.

-π 0 π x

1

PHY226

Phil Lightfoot 2008/9 Tutorial Questions - Page 7 of 10

7. (a) Show that the function shown has Fourier series

−+−+= ...5cos

513cos

31cos2

21)( xxxxf

π

−π -π/2 0 π/2 π x

PHY226

Phil Lightfoot 2008/9 Tutorial Questions - Page 8 of 10

(b) Hence deduce that the function below has series

−+−+= ...10cos

516cos

312cos2

2)(

Tt

Tt

Ttaatf πππ

π

−Τ/2 -Τ/4 0 Τ/4 Τ/2 t

a

PHY226

Phil Lightfoot 2008/9 Tutorial Questions - Page 9 of 10

8. A function is defined on the range [0, π]: f(x) = x 0 < x < π

(a) Sketch the odd extension of this function. Show that the corresponding half-range sine series is

∑∞

=

+−=

1

1

sin)1(2)(n

n

nxn

xf

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Phil Lightfoot 2008/9 Tutorial Questions - Page 10 of 10

(b) Sketch the even extension of this function. Find the corresponding half-range cosine series.