Computing Across the Physics Curriculum
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Mechanics
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These modules were designed/constructed based on the text:
Classical Mechanics, by R. Taylor.
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Coupled Pendulum 1st Day Demo: The demonstration is used to engage students with a complex physical system unlike anything they have seen in the introductory physics course. The system is a coupled oscillator whose motion can be described by two coupled second order differential equations. The equations are derived using Lagrangian Mechanics, which is covered at the end of the course. So this demonstration on the first day gives students an idea of where we are headed in the class. The demonstration also introduces the technique of using computer simulations to describe the motion of complex systems - a technique that is used throughout the course.
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Simulating Physics Systems Using Euler and ODE45: The simple example of an apple falling from a tree is simulated using both Euler and ODE45 methods. This example is used because the analytic solution is familiar to students.
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Coupled Pendulum Simulation Homework: This is a homework assignment to follow up on the 1st day demonstration of the coupled pendulum. The assignment is designed to give students individual experience exploring the simulation introduced in class. Students use MATLAB code provided by instructor to answer the question, "Does the period of oscillation of the oscillating mass change as the length of the string increases? Provide a plot from MATLAB to justify your answer".
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Modified Euler: Students explore the error that in introduced with Euler's Method with a harmonic oscillator (mass on spring) and explore ways to minimize the error. This is an important assignment as it provides students with insights into the types of artifacts that can enter into numerical simulations. This material should be review since it is covered in our introductory physics computation course, but there may be transfer students who have not had our computation course and it is not a bad idea to review these ideas since they are so ubiquitous in this course.
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Plotting Force to Find Minimum Angle: A dog is pulled along a surface at constant speed with friction. The force required to get the dog to slip depends on the angle and the coefficient of friction. Students find the angle as a function of ìsing calculus. They then solve the problem by simply plotting the quantity that is maximized and reading the angle from their plot. This problem was solved easily using either method, but in many cases in real life, an analytic solution can be difficult or impossible to obtain - so learning how to solve problems with graphical methods is an essential skill for our students.
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Horizontal Motion with Linear Drag: In this lecture, the equation of motion is found analytically by solving a second order linear differential equation. Euler's method in MATLAB is then presented as an independent way to check the validity of the analytic solution.
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Finding Maximum Range of Projectile Shot from Mountain: In this homework problem, students are provided MATLAB code to find the maximum range of a projectile shot from an elevated position. This problem was done as an example from lecture. Students are then asked to modify the program to investigate a slightly different problem, that is: "How does maximum range of a projectile shot from an elevated position vary as a function of launch height". It is obvious that the maximum range should increase as launch height increases, but it is not a simple linear relationship.
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Simultaneous Equations from P and E Conservation: When we have both momentum and energy conservation, we end up having to solve simultaneous equations. This problem is always done in intro courses, but students usually don't solve the equations themselves.
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Exploring Rocket Motion: The iterative equations for momentum conservation for a baby throwing grains of sand from a frictionless wagon were worked out in L3-2. The wagon/sand system was meant as an analog to a rocket that exhausts fuel mass. After the L3-2 lecture, students generally have no idea how the final velocity of the a rocket depends on the relative mass of the fuel and exhaust speed. This assignment was developed to give students the opportunity to explore the rocket/fuel system and explore how the motion graphs and final rocket velocity depend on the fuel exhaust velocity, rate of fuel loss, and relative masses of rocket and fuel. Ideally, students develop an appreciation for these relationships before the next lecture where the relationships are worked out analytically. Students are provided the code, but they must include the critical Euler approximation line to complete the program. The basic equation was derived in lecture, but students must adapt the equation to match the program terms.
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Predicting Motion of Particle in a 1D Double-well Potential: There are many characteristics of the motion of a particle in a 1-D potential well that can be easily determined by examining the potential function. By observing the potential energy function arising from a conservative force function, students can determine the characteristic motion of a particle in the well. The ability to predict the motion of a particle in a 1 dimensional potential well not just critical to mechanics, but also to many other branches of physics including quantum mechanics, fluid mechanics, celestial mechanics, electricity and magnetism, chemical physics, nuclear physics and others.
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Oscillating Bead on a Bent Wire: Finding the equation of motion of a bead released on a bent wire provides a wonderful example of a non-trivial curvilinear 1D system whose equation of motion is susceptible to the conservation of energy approach. This problem is nice because the conservation of energy equation is very straightforward to write and is familiar to students from their first introductory physics course. In the introductory physics course, students were able to find velocity as a function of position using energy techniques, but not position or velocity as a function of time. We take the problem to a higher level in our Theoretical Mechanics course by solving for the velocity and solving the differential equation to find the equation of motion.
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Bead on A Parabolic Wire: This is a major homework assignment that takes students quite a while to complete. The assignment walks them through writing the conservation of energy equation for a bead on a parabolic wire. They then develop the differential equation governing the motion of the bead and write a simulation for the system. They use the MATLAB code from L4-2 as the basis of their program, so writing the code is not a challenge.
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Falling Stick: A stick leans against a wall. Students use Lagrange's approach to generate a differential equation for the stick's motion as it falls. Students use numerical integration to obtain the actual time it takes for a stick to fall.
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Additional MATLAB Applications for Theoretical Mechanics. Dr. James has many more Matlab programs to share. This PDF file contains descriptions and contact info.
