SOLUTIONS MANUAL FOR
COMPUTATIONAL
MATERIALS SCIENCE
An Introduction
by
June Gunn Lee
** Immediate Download
** Swift Response
** All Chapters included
, Homework and Solutions
Table of Contents
Chapter 2 01
Chapter 3 07
Chapter 4 20
Chapter 5 25
Chapter 6 29
Chapter 7 33
Chapter 8 37
Chapter 2
Homework 2.1
Consider a one-dimensional harmonic oscillator of mass, m, and spring constant of k that follows
Hooke's law, F = −kx , in an isolated system (E = const.). Then the Newton's second law
becomes the second-order ordinary differential equation as:
k
a (t ) = − x(t )
m
Let us assume that the spring is fixed at one side and is ideal with zero mass. Given oscillator
=
frequency of ω (=
k m)1/2 1 and initial conditions of x0 = 5 and v0 = 0 , find and discuss the
following:
• Write down the potential expression for the system
• Analytical solution for ball positions with time
1
, • Numerical solution for ball positions with time by taking the first two terms in the
Taylor series expansion (the first-order Euler method) with ∆t = 0.2 for 10 timesteps
• Compare the two methods above on a position-time plot and discuss what will happen if
the numerical calculation proceeds further for more timesteps and how to improve its
accuracy
Solution 2.1
Model
An ideal ball-spring system in one dimension.
Potential expression
F = −kx
1
U =− ∫ Fdx =− ∫ −kxdx = kx2 : Parabolic curve
2
2
, Analytical solution:
F = −kx= ma , 0 = −kx= ma at equilibrium
This second-order differential equation has the general solutions as:
v0 cos(ωt ) − ω x0 sin(ωt ) =
v(t ) = −5sin(t )
v0
x(t ) = x0 cos(ωt ) + sin(ωt ) = 5cos(t )
ω
Numerical solution:
v(t + ∆=
t ) v(0) + a (0)∆t , …..
x(t + ∆=
t ) x(0) + v(0)∆t , …..
−kx= ma , k m = 1 → a = − x
Numerical calculation for one-dimensional harmonic oscillator
t ∆t x a = −x v
0.0 - x0 = 5 a 0 = -5 v0 = 0
x1 = x0 + v0∆t v1 = v0 + a 0∆t
0.2 0.2 a 1 = -5
=5+0=5 = 0 - 1 = -1
0.4 0.2 x2 = x1 + v1∆t a 2 = -4.8 v2 = v1 + a 1∆t
3
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller mizhouubcca. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $29.49. You're not tied to anything after your purchase.
