Numerical Approximation
 You have some Physics equation or equations
which need to be solved
 But:
• You can’t or don’t want to do all that
mathematics, or
• The equations can not be solved
What to do?
Numerical Approximation
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Numerical Approximation Module
 Based on the Python programming language
and the Visual Python package VPython
 We will investigate a system that you will soon
be exploring in some detail in the course: the
oscillating spring-mass system
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About Python and VPython
 An ideal 1st programming language
 Not a toy: used for production programs by
Google, YouTube, etc.
 Open source
 Traditionally for all languages, for beginners the
first “program” only prints:
hello, world
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Here is a complete Python program
that prints: hello, world
print ”hello, world”
Note the quotes
Totally intolerant of typing mistakes: this will not work
prind ”hello, world”
Case sensitive: this won’t work either
Print ”hello, world”
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The VPython environment
Here is a VPython window ready to run our first
program
To run the program, click on Run and choose
Run Module
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A second window will appear:
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Another complete Python program
that prints hello, world
A variable is given the
named what
value
world
what = ”world”
print ”hello,”, what
First Python executes the first line of the program
Next Python executes the second line of the
program: it prints hello, followed by the value of the
variable what
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Loops
Often we wish to have a program execute the
same lines over and over
Loops do this
Assign variable x a value of 0
Example:
Is x less than 3?
If so, execute the
x = 0
following lines of
while x < 3:
program. If not, stop
print x
Increase the value of
x = x + 1
x by 1. Go back to the
while statement
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The Spring-Mass System
The force exerted on the mass by the spring:
F = -k x
(Hooke’s Law)
F=ma
(Newton’s Second Law)
Combine to form a
differential equation:
2
d x
ma  m 2  kx
dt
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Solving Differential Equations
1. Learn the math, or
2. Find a mathematician, or
2
d x
m 2  kx
dt
3. Get hold of software that can solve differential
equations, such as Maple or Mathematica
If you choose #2, note that you don’t need to tell
them what, if anything, the equation is about
Solving differential equations has nothing to do
with Physics!
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The Mathematical Solution
x  A sin( t )
2
d x
m 2  kx
dt
k

m
You will be learning about this soon in class
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Avoiding all that mathematics
Recall: ma = -kx
At some time t we know the position x of the
mass and its speed v
1. Calculate the acceleration a = - (k/m) x
2. Calculate its speed a small time Dt later:
vnew = v + a Dt
3. Calculate its position a small time Dt later:
xnew = x + vnew Dt
Go back to #1 and repeat over and over
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Avoiding all that mathematics continued
1. Calculate the acceleration a = - (k/m) x
2. Calculate its speed a small time Dt later:
vnew = v + a Dt
3. Calculate its position a small time Dt later:
xnew = x + vnew Dt
Go back to #1 and repeat over and over.
This method is “numerical approximation”
This can be made as close to correct as we
desire by making the “time step” Dt
sufficiently small
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What you will do today

We have prepared a VPython program that
animates the mass of a spring-mass system
two different ways:
1. By coding the solution to the differential
equation
2. By numerical approximation
• You will examine the code and identify which
parts do what
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