MATHEMATICA – Computer Simulation R.C. Verma Physics Department Punjabi University

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MATHEMATICA – Computer Simulation R.C. Verma Physics Department Punjabi University

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MATHEMATICA – Computer Simulation R.C. Verma Physics Department Punjabi University

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MATHEMATICA – Computer Simulation

R.C. Verma

Physics Department

Punjabi University

Patiala – 147 002

PART IX- Computer Simulation

Mechanics

- Traditionally physics teaching comprises of theory lectures based on analytical techniques and conventional laboratory experiments.
- Despite the importance of computational physics, it has been largely neglected in the conventional physics curricula.
- Now with the availability of personal computers, it has become possible to introduce this important branch in the physics curricula.

- PC offers new opportunities for innovative learning.
- It provides highly interactive, individual and creative learning.
- It can help to approach wide variety of problems and phenomena than is possible with only analytic tools.
- It can also be used to develop physical intuition and ability to estimate physical quantities involved in a phenomena.
- NMEICT (MHRD, Govt. of India)- Rs. 4,612 crores Mission

For practical purposes of PC into physics we need to answer:-

- How to use PC to improve physics teaching?
- What other changes will come after we introduce PC to the physics curricula?
- Could advances of research into physics learning be incorporated into new Curricula?
- Can the new curricula reflect contemporary physics?

i) Number awareness

ii) Experimental skills

iii) Analytic skills

iv) Scales and estimations

v) Approximations skills

vi) Numerical skills

vii) Intuition & large problem skills

- Problem Solving
- Demonstrations and Tutorials (CAI)
- Data analysis using Spreadsheets
- Simulation of Physics Problems
- Graphics and Animation
- Magnification of Instruction

- PC can be used easily and interactively through a variety of high-level languages,
- They offer numerical power sufficient for even initiating research-level problems.
- Many numerical programming languages are already with us: BASIC, FORTRAN, and C
- Recently, Symbolic Computational languages: Mathematica, MatLab, MathCad, Macsyma
- capable of dealing with algebra, differential and integral calculus, and powerful graphics tools.
- This obviously enhances the scope of physics problems to be handled on a PC.

- The corner stone of computing is building a model of an idea through simulation.
- It can deliver real time sequence on the screen.
- We can simulate real world phenomena that are prevented from studying in the laboratory due to constraints of time, expense, danger and feasibility.
- E.g.Planetary Motion, Nuclear Reactor, Interior of Sun
- We can try models that don't occur in real world to seewhat the implication would be.
- E.g. What would happen if we change the gravitational force law a little?

- Computational physics has largely been neglected in the standard physics curricula.
- Main factors : 1. Lack of computing hardware 2. Lack of teaching-material besides 3. Lack of trained human resource.
- Situation is slowly improving.

Physics → Algorithm → Program → Results

- Computer Hardware and Software
- Numerical analysis
- Development of algorithms for problems
- Developments of programs for simulation
- Results and Error analysis.

Identify the input variables:-like parameters of a physical systeminitial conditions of the`system and time interval step size of time evolution.

Identify the output variables:-solution of the problem.

Construct the equations to connect the input variables to the output variables.

Re-express the equations using numerical techniques.

Write algorithm/flowchart to solve the problem.

Develop Programs (I/O, common arithmetic operations and logical structures: Sequential, Repetitive and Selective).

Execute the program on a computer.

Run computer experiments to study effects of:

- change of step size used in discretization of continuous independent variable;
- change of initial conditions of the physical system;
- change of physical parameters of the system.
- changes due to errors,
- stability and limitations of the numerical tools.

- A spherical body falling in viscous medium

- Equation of motion is

Clear["Global`*"]

(* Find Analytic solution *)

k = 3.0;(* spring constant *)

m = 1.0;(* mass attached to the spring *)

w0 = Sqrt[k/m];

c = 0.5;

damp = c/m;

x0=1.0; v0 = 1.0;(* initial conditions *)

tmin = 0;tmax = 5;

ndsol=DSolve[ {x''[t]+damp*x'[t]+w0^2 x[t]==0,

x[0]==x0, x'[0]==v0}, x[t], t]//Chop//Flatten

(* Plot the solution for a given time interval *)p1= Plot[ x[t]/.ndsol, {t,tmin, tmax}, AxesLabel->{"t->", "x"}, PlotLabel->"Harmonic Motion"]