My approach to undergraduate Mechanics is to view Newton’s Laws as a holistic set, which allows us to predict the behavior of a physical system, starting with some initial configuration, into the future and indeed, into the past (the inherent microscopic time-reversibility in Newtonian Physics). A lot of emphasis is placed on using these laws to try to solve ‘real’ problems, rather than just textbook problems. In this context, I introduce my students to the “Spherical Cow”
https://en.wikipedia.org/wiki/Spherical_cow
https://en.wikipedia.org/wiki/Spherical_cow
The idea is to use successive approximations to describe the physical system, till the predictions are consistent with experiments (as much as possible). That is, give the spherical cow an increasingly cow-like structure
- Lectures on Mechanics
- Newton’s Laws on a Computer
- Dimensional Analysis
- The Pendulum
- Planetary Motion
- Computational Projects
Lectures on Mechanics
Following are lecture slides I have often used in the classroom. These are just an outline, supplementing blackboard work. Conspicuous is the absence of lectures on Relativity. Those are still a work in progress. There could be quite a few typos and /or logical errors. These are usually avoided in the classroom.
Classical_Mechanics_Lecture_SlidesNewton’s Laws on a Computer
Having developed some intuition for Newton’s Laws, it is time to use them to model ‘real’ physical systems, using the Spherical Cow paradigm. This is usually non-trivial, so I introduce ‘Computational Projects’ to students so that they get a feel for this. I begin by introducing algorithms to implement Newton’s Laws on the computer
Newtons_Laws_AlgorithmsDimensional Analysis
In the classroom, I place a lot of emphasis on the use of dimensional analysis, both as a philosophy and as a practical tool for modelling physical systems on a computer. As a philosophy, it relates ‘constants’ appearing in Newton’s Laws for a physical system to inherent ‘scales’ in the system, identifying which gives intuition about the system even before solving the (often complex) set of differential equations. At a deeper level, the fundamental constants of nature give us deep insight into the structure of physical theories and their scale dependence. I often introduce students to the ‘Cube of Physics’ (not an original insight, but borrowed from https://cube-of-physics.org/)
For instance, the introduction of Planck’s constant to electrostatics generates a natural length scale (the ‘size’ of a atom), pointing to the existence of Quantum Theory. The fundamental constants of nature (Planck’s constant, Newton’s Gravitational constant and the speed of light in vacuum) point to the existence of the Planck Scale, etc. At a purely practical level, dimensional analysis allows us to use the algorithms for solving Newton’s Laws in a sensible way, without having to worry about encountering very large or small numbers. Reducing equations to dimensionless form before solving them is extremely important, since a computer does not understand dimensions, but pure numbers. Following are python programs simulating motion of a pendulum and an object in presence of gravitation. These jupyter notebooks discuss the dimensional analysis relevant to solving the problems, and why it is so important. These also introduce the students to the various algorithms.
The Pendulum
The following code simulates the motion of a pendulum (for arbitrary initial displacement). Euler’s algorithm is introduced, and upon observing that the error cannot possibly be tolerated, the much more accurate Verlet algorithm is discussed. When damping is introduced, the Verlet algorithm does not work (works only for conservative systems). So, the second order Runge-Kutta algorithm is invoked
Planetary Motion
Motion of a projectile in the vicinity of a spherically symmetric gravitating object is simulated. The initial velocity of the projectile is the control variable. Changing this generated different orbits. The Verlet algorithm is used.
Computational Projects
Following are a few computational ‘projects’ the students are encouraged to work on. The idea is to expose them to modelling ‘real’ physical systems through successive approximations, eventually making the models too complicated to solve by pen and paper. I have been offering these projects for a few years now, and they seem a little dated (at least to me, if not to the students). Following are the most recent versions offered this semester to first year undergrads
Chaotic Moon
The rotational motion of an irregular moon of a planet in orbit around a star can be chaotic (example: motion of Hyperion, a satellite of Saturn: https://www.jpl.nasa.gov/images/pia05433-tumbling-hyperion). The following computational exercise breaks down the problem into smaller, simpler chunks and encourages the students to demonstrate chaos in the system.
Chaotic_MoonPhysics of Golf
The trajectory of a golf ball in presence of air drag and the Magnus Effect. There are quite a few surprises here. Above all, this project is an excellent illustration of transforming the ‘Spherical Cow’ into a ‘real’ animal.
Golf_PhysicsKirkwood Gaps
‘Mysterious’ gaps in the asteroid belt between Mars and Jupiter. What causes them?
Jupiter_AsteroidsPhysics of Raindrops
The very interesting Physics of the change in shape and velocity of raindrops as they fall. Another excellent illustration of the ‘Spherical Cow’ paradigm.
Raindrop_Physics