Classical Mechanics

Spring 2025 Review in Tweets

Lately in #PHY471 #ClassicalMechanics: Using Newton's 2nd law, we derived the equations describing a projectile, (1) without air resistance, and (2) with linear drag. This involved solving lots of 1st order ordinary differential equations using separation of variables. pic.twitter.com/TLR8rUR2Gp

— Tom Wong (@thomasgwong) January 28, 2025

Yesterday in #PHY471 #ClassicalMechanics: We derived the trajectory of a projectile under linear drag, and we plotted it for various drag coefficients. For many students, this was their first time using Mathematica, may be just as important as learning the physics! pic.twitter.com/opr7mmQmpZ

— Tom Wong (@thomasgwong) January 30, 2025

Lately in #PHY471 #ClassicalMechanics: My students have been learning a very valuable lesson: Nature is under no obligation to be described by equations we can actually solve. In such cases, we can (1) numerically solve them, or (2) solve them approximately or for limited cases.

— Tom Wong (@thomasgwong) February 6, 2025

Last Friday in #PHY471 #ClassicalMechanics: We reviewed some Newton's 2nd law problems from general physics. Then, we solved a block sliding down a wedge that can slide on a table, which involved solving a system of five equations and five unknowns in Mathematica. pic.twitter.com/Wqnq3CDakq

— Tom Wong (@thomasgwong) February 10, 2025

Yesterday in #PHY471 #ClassicalMechanics: In Cartesian coordinates, Newton's 2nd law is equivalent to 1D versions of the same law. In polar coordinates, it's more complicated.

Lagrangian and Hamiltonian mechanics will handle other coordinate systems as easily as Cartesian! pic.twitter.com/nBultMATwf

— Tom Wong (@thomasgwong) February 11, 2025

For Valentine's Day, I gave my #PHY471 #ClassicalMechanics students a midterm. It covered projectile motion with linear and quadratic drag, and Newton's 2nd Law in Cartesian and polar coordinates. I hope they felt the love...

— Tom Wong (@thomasgwong) February 14, 2025

Last time in #PHY471 #ClassicalMechanics: The work done by a force on an object is obtained by adding up the infinitesimal displacements of the object's path, times the force along each infinitesimal displacement. pic.twitter.com/scyuw1s893

— Tom Wong (@thomasgwong) February 19, 2025

Today in #PHY471 #ClassicalMechanics: The total work on an object equals its change in kinetic energy. Potential energy is negative work, and force is the negative gradient of potential energy. The change in mechanical energy is equal to the work done by nonconservative forces. pic.twitter.com/OYulrEhcPT

— Tom Wong (@thomasgwong) February 19, 2025

Yesterday in #PHY471 #ClassicalMechanics: We reviewed cross products and the curl of a vector. A force is conservative (and hence has a corresponding potential energy) if and only if its curl is zero. pic.twitter.com/o3pzaxjyXM

— Tom Wong (@thomasgwong) February 23, 2025

Today in #PHY471 #ClassicalMechanics: Momentum, conservation of momentum, and impulse. We derived the rocket equation! pic.twitter.com/EdrVMBQBhr

— Tom Wong (@thomasgwong) February 24, 2025

Lately in #PHY471 #ClassicalMechanics: Rotation, torque, angular momentum, moment of inertia, parallel axis theorem, conservation of angular momentum, and the inertia tensor. pic.twitter.com/RV8uou9nL4

— Tom Wong (@thomasgwong) March 3, 2025

Today in #PHY471 #ClassicalMechanics: Calculus of variations and the brachistochrone problem, which is to find the path for a frictionless roller coaster to reach its end point in the shortest time. The answer is a cycloid, which is the path traced by a point on a rolling circle. pic.twitter.com/G9UlmYuHqW

— Tom Wong (@thomasgwong) March 17, 2025

Yesterday in #PHY471 #ClassicalMechanics: We used Lagrangian mechanics to solve the problem of a bead sliding on a rotating hoop. The equilibrium at the bottom is stable for slow rotations and unstable for fast, but fast rotation results in another stable equilibrium to the side. pic.twitter.com/t8irJ736Zz

— Tom Wong (@thomasgwong) March 25, 2025

Yesterday in #PHY471 #ClassicalMechanics: Examples of Lagrangian mechanics, including a sphere rolling down a ramp, the Atwood machine, a pendulum whose anchor is accelerating to the right (animation below), and a block sliding down a wedge that slides on a table. pic.twitter.com/hfQ9mwra97

— Tom Wong (@thomasgwong) March 27, 2025

Today in #PHY471 #ClassicalMechanics: We began a new unit on Gravity and Orbits, starting with Newton's Law of Universal Gravitation and the Shell Theorem. Here's a relevant Physics GRE problem: pic.twitter.com/bQFwwiEhnc

— Tom Wong (@thomasgwong) March 31, 2025

Today in #PHY471 #ClassicalMechanics: We derived the equation of an orbit around the sun.
Picture 1: The orbit equation appears in the movie, Hidden Figures.
Picture 2: The orbit is a circle, ellipse, parabola, or hyperbola, depending on the eccentricity/energy. pic.twitter.com/AGKVyOW0Jz

— Tom Wong (@thomasgwong) April 4, 2025

Last week in #PHY471 #ClassicalMechanics: The third midterm, covering calculus of variations, Lagrangian mechanics, and gravity and orbits. pic.twitter.com/ndBxaTwjrJ

— Tom Wong (@thomasgwong) April 15, 2025

Yesterday in #PHY471 #ClassicalMechanics: Hamiltonian mechanics. The Hamiltonian is a function of position and momentum, and it's conserved for time-invariant systems. The rate of change of position and momentum is given by partial derivatives of the Hamiltonian. pic.twitter.com/RxxbnGsgld

— Tom Wong (@thomasgwong) April 15, 2025

Today in #PHY471 #ClassicalMechanics: Examples of Hamiltonian mechanics, phase space orbits, and Liouville's theorem. Here's a particle moving in a cone. pic.twitter.com/W6KIXh2MMP

— Tom Wong (@thomasgwong) April 16, 2025

Yesterday in #PHY471 #ClassicalMechanics: Parallel and series springs exhibit simple harmonic motion with effective spring constants, but coupled springs are only sinusoidal (called normal modes) at certain frequencies (called normal frequencies). pic.twitter.com/8GEJYtozfQ

— Tom Wong (@thomasgwong) April 24, 2025

Today in #PHY471 #ClassicalMechanics: The simple pendulum and physical pendulum exhibit simple harmonic motion for small angles. The double pendulum is chaotic, so a small change in initial conditions results in dramatically different behavior. pic.twitter.com/RZCr5KSHdX

— Tom Wong (@thomasgwong) April 26, 2025

Today in #PHY471 #ClassicalMechanics: Any periodic function can be expressed as a sum of sines and cosines, called a Fourier series. Pictured is the truncated Fourier series that approximates a periodic rectangular pulse as more terms are kept. pic.twitter.com/p4tsfSYUhS

— Tom Wong (@thomasgwong) April 30, 2025