Numerical Approximation 1

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: theoscillating 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 programthat 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

To run the program, click on Run and chooseRun Module

Here is a VPython window ready to run our first program

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A second window will appear:

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Another complete Python programthat prints hello, world

what = ”world”print ”hello,”, what

First Python executes the first line of the program

A variable named what

is given thevalue

world

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

Example:

x = 0while x < 3:

print xx = x + 1

Assign variable x a value of 0

Is x less than 3? If so, execute the following lines of program. If not, stop

Increase the value ofx by 1. Go back to thewhile 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 = m a (Newton’s Second Law)

kxdt

xdmma

2

2Combine to form a

differential equation:

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Solving Differential Equations

kxdt

xdm

2

2 1. Learn the math, or

2. Find a mathematician, or

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 dowith Physics!

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The Mathematical Solution

kxdt

xdm

2

2

)sin( tAx

m

k

You will be learning about this soon in class

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Avoiding all that mathematics

1. Calculate the acceleration a = - (k/m) x

2. Calculate its speed a small time t later:vnew = v + a t

3. Calculate its position a small time t later:xnew = x + vnew t

Recall: ma = -kx

At some time t we know the position x of the mass and its speed v

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 t later:vnew = v + a t

3. Calculate its position a small time t later:xnew = x + vnew t

Go back to #1 and repeat over and over.

This can be made as close to correct as we desire by making the “time step” t sufficiently small

This method is “numerical approximation”

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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