physics-problems

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A Thought-Provoking Toy

by Michael Keller

The spinning top above illustrates an unusual asymmetry where it flips over if spun in a clockwise motion and stays upright when spun counterclockwise. This behavior is a result of chirality, a property in which something displays handedness. When an object or system is chiral, its mirror images can’t be exactly mapped to each other–like your right and left hands. 

Tadashi Tokieda, director of studies in mathematics at Trinity Hall, University of Cambridge, investigates and invents toys like the one above that exhibit interesting behaviors. He’s also a fellow at Harvard’s Radcliffe Institute for Advanced Study, where he presented what he calls the world’s first chiral tippy top. See the video with this and other toys that display chirality below.

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The stages of AP physics homework:
  • Getting out book:I totally got it this time! I actually understood the notes today
  • Looking at problem:wow that's a long problem, but I can do it.
  • Reading problem:what does that even mean?
  • Trying problem:ill just write all the variables and wing it
  • :we never learned this
  • :how do I do this
  • :*cries*
  • :throws book across room
  • :realizes I need the book
  • :gets book
  • :cries more
  • :draws someone jumping off a cliff and label It physics
  • :gives up
  • :cries more as eats out of sadness
2

Solar Coronal Waves

One of the most longstanding question in solar physics is why the Sun’s corona is hotter than its surface. This is known as the ’coronal heating problem’, and there’s no definite answer to that question. The Sun is a very complex magnetohydrodynamic system; there are different kinds of flows and instabilities that can play an important role in the energy-transport processes in the corona. The waves – pictured above, are initiated by a Kelvin-Helmholtz instability. The Kelvin–Helmholtz instability occurs not only in clouds and in the ocean, but also in various astrophysical environments. This instability occurs when two flows of different velocities move past one another in opposite directions with a strong enough shear to overcome the tension force. In the solar atmosphere, which is made of hot ionized plasma, the interplay of two hot-plasma jets and the embedded magnetic field might trigger turbulent flows which could help add heating energy to the corona.

For more information:

Credit: NASA/JPL/GSFC

Reports emerged this weekend that Kazakh mathematician Mukhtarbay Otelbaev has published a proposed solution to the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize problems offered by the Clay Mathematics Institute. Today I want to explain some of the background of this problem, what is known about Otelbaev’s proposed solution, and what a solution would mean for fluid dynamics.

The Navier Stokes Equation

The Navier-Stokes equation is one of the governing equations of fluid dynamics and is an expression of conservation of momentum in a fluid. With the exception of a few very specific and simplified cases, there is no known general solution to equation. Instead, the equation, or a simplified model, is solved numerically using supercomputers as part of direct numerical simulation (DNS) or other forms of computational fluid dynamics (CFD). These methods allow scientists and engineers to solve the equations of fluid motion for practical problems from flow through a pipe to flow around a re-entering spacecraft.

Existence and Smoothness

Although the Navier-Stokes equation has been known for more than 150 years and can be solved numerically for many situations, some basic mathematical aspects of the equation have not yet been proven. For example, no one has proven that a general solution always exists in three-dimensions and that the energy of such a solution is bounded at all points. Colloquially, this is known as the Navier-Stokes existence and smoothness problem. The Clay Mathematics Institute has a very specific problem statement (PDF) asking for a proof (or counter-proof) of the existence and smoothness of the Navier-Stokes equation for an incompressible fluid in three-dimensions. Otelbaev contends that he has provided such a proof.

Otelbaev’s Proposed Solution

Mukhtarbay Otelbaev is an experienced mathematician with numerous papers addressing related mathematical problems. His latest paper, entitled “Existence of a strong solution to the Navier-Stokes equation,” is freely available online (PDF, in Russian, with an English abstract at the end). There is an ongoing project to translate the paper into English, and mathematicians are already evaluating the validity of this proposed solution. From what I can gather of the paper, it specifically address the Millennium Prize problem and presents Otelbaev’s proposed solution for the existence and smoothness of an incompressible fluid in three dimensions with periodic boundary conditions.

What It Means

As with any announcement of a major technical breakthrough, skepticism is warranted while experts evaluate the proposal. If the mathematical community upholds the validity of Otelbaev’s proof, he may be offered the Millennium Prize and other honors. More importantly, his solution could lead to a better understanding of the nature of the equation and the flows it describes. It is not, in itself, a general solution to the Navier-Stokes equation, but it may be a stepping stone in the path toward one. In the meantime, scientists and engineers will continue to rely on a combination of theory, experiment, and computation to progress our understanding of fluid dynamics.

For More

The story of Otelbaev’s proof and the community’s evaluation of its validity is on-going. You can follow @fyfluiddynamics and the #NavierStokes hashtag on Twitter for updates and commentary. I’d like to specially thank Catriona Stokes, Praveen C, David Sarma, and Glenn Carlson for their helpful links and observations as this story develops.

I think somewhere between you being an electron and me being a proton, I fall in love with you so madly. We are electric, magnetic, and dynamic. You are attracted to the way I pull you off me, and nothing would kill you but the electroshock running through your veins every time you want to touch me.

And you put my molecules on fire and they cry while burning: I am in love, I am in love and I love you. Moreover, I pretend I don’t hear them cause it hurts too much to know that we are not a symbiosis, that you only take and take and never ever give anything back.

Why don’t you combine your cells with mine? Destroy the walls and let them float in this blood of madness. Because we are not from here and every element of us is calling towards the west coast, and why don’t you die on my heart? Let the scientists study who we are…

Because we are a different kind of species.

—  Science Is Writing About Us, Baby by Royla Asghar
How to Study Physics Like a Pro

we’ve all been there, looking at a physics text book and thinking: “wtf is going on here???”. so here is a simple and intuitive guide on how to understand and apply this subject.

needless to say, first of all understand the theory

  • seek the general theory behind the topic that you’re studying - this will give you a main equation which the others are all related to
  • you need to know at least the basics of math, like calculus and algebra. focus on solving equations
  • all the theories of a particular topic are connected - seek the similarities and connections
  • memorize the main equations and extract the others from them - this will save you memory and time
  • visualize the phenomena in your head
  • if you still have doubts, ask. ask your teacher to explain it

prepare yourself in advance

  • don’t procrastinate!
  • pay attention in class and take notes
  • revise, revise and revise.
  • exercise, exercise and exercise
  • at this point it’s all about practice - more often than not when you understand to solve a kind of problem you can solve all the problems of that kind
  • seriously, practice makes perfect
  • breathe and stay focused 

problem solving

  • read the question and analyze it
  • pick all the useful information and write it down
  • draw the phenomena
  • take a step by step approach
  • what needs to be solved?
  • how are the phenomena related?
  • fit the data in the most appropriate equation
  • does the solution make sense?
  • done!
  • if you have time at the end of the test check carefully if you’ve done any  mistake

this could be helpful

youtube

Mind-blowing quiz of the week.

Is 1 + 2 + 3 + 4 + 5 + … = -1/12?

The answer depends a lot on if you grab the problem from a mathematical point of view, or from the physical one. Both are legitimate and useful in its application framework.

More info in “When Is Actually a Small Infinity, Negative Fraction” and “Follow-up: The Infinite Series and the Mind-Blowing Result” (Bad Astronomy).

And an in-depth (and heterodox) explanation also at The Reference Frame: “Sum of integers and oversold common sense”.

Enjoy, if you can.

I always thought those physics word problems were so stupid like “You’re a physics student so you decided to figure out the speed of ____ from the variables you’re given like the nerdy asshole you are” and I was always like “literally no one is like that” but today I was in the hall elevator and it’s always so fucking slow and so I was like “well if I measure the time it takes to go from one floor to the next while it’s at top velocity and I take the average length of a floor in my building I can figure out that this elevator travels at 2.3 m/s or 5.1 mph and then I can even take the average time it takes a person to climb a flight of stairs to figure out if this elevator is faster than the stairs or not IVE BECOME THAT NERDY ASSHOLE IN THOSE STORY PROBLEMS I HAVE BECOME THE ENEMY