Different modes of oscillation for a pendulum
The period of a simple pendulum is not a trivial thing, and it depends on the initial conditions.
Shown here are ten different modes of oscillation for the same pendulum. The only difference is the total amount of mechanical energy in the system.
As a result, each one has a completely different period of oscillation, unlike what the small-angle approximation (as taught in high-school) would suggest. They can’t be in sync. You may see some really interesting patterns based on the delay between them in your browser.
The red graph above each pendulum represents the phase portrait for the respective mode of oscillation, with the current state marked as a blue dot. The horizontal axis represents angle (hence why it wraps around the sides) while the vertical axis represents angular velocity.
Pendulums are very interesting dynamical systems, as they are relatively simple to understand but can produce surprisingly complex results in certain cases, such as the chaotic behavior of double pendulums and the odd behavior displayed by coupled pendulums.
math homework is 10% thinking and 90% tears
Science Day in India, posting whole series of Scientists, their inventions or discoveries.
The familiar trigonometric functions can be geometrically derived from a circle.
But what if, instead of the circle, we used a regular polygon?
In this animation, we see what the “polygonal sine” looks like for the square and the hexagon. The polygon is such that the inscribed circle has radius 1.
We’ll keep using the angle from the x-axis as the function’s input, instead of the distance along the shape’s boundary. (These are only the same value in the case of a unit circle!) This is why the square does not trace a straight diagonal line, as you might expect, but a segment of the tangent function. In other words, the speed of the dot around the polygon is not constant anymore, but the angle the dot makes changes at a constant rate.
Since these polygons are not perfectly symmetrical like the circle, the function will depend on the orientation of the polygon.
More on this subject and derivations of the functions can be found in this other post
Now you can also listen to what these waves sound like.
This technique is general for any polar curve. Here’s a heart’s sine function, for instance
The technique used by Brand to create these pieces is not one of conventional holography. He meticulously controls the unique shape of thousands of tiny optical pieces placed on a surface creating a 3D effect when the light source or viewer moves. This is essentially a mathematical problem in differential geometry and combinatorial optimization. Brand was the first person to correctly describe this technique in 2008 even though it dates back as early as the 1930s (check out his paper for details).
A newly found asteroid will pass just inside the orbit of the Moon, with its closest approach on March 4, 2013 at 07:35 UTC. Named 2013 EC, the asteroid is about the size of the space rock that exploded over Russia two and a half weeks ago, somewhere between 10-17 meters wide (the Russian meteorite is estimated to be about 15 meters wide when it entered Earth’s atmosphere). 2014 EC was discovered by the Mt. Lemmon Observatory in Arizona on March 2. . There is no chance this asteroid will hit Earth.
2013 EC will come within 396,000 kilometers from Earth, (246,000 miles, or around 1.0 lunar distances, 0.0026 AU.
The Moon’s distance from the Earth varies between 363,104 km (225,622 miles) at perigee (closest) and 406,696 km (252,088 miles) at apogee (most distant point).
Gianluca Masi from the Virtual Telescope Project had a live view of the asteroid when it was about twice the distance of the Moon, and a replay of that webcast is available below.
“That we are finding all these asteroids recently does not mean that we are being visited by more asteroids,” Masi said during the webcast, “just that our ability to detect them has gotten so much better. Our technology has improved a lot over the past decades.”
J. R. Eyerman - Space Frontiers (c. 1961)