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Lagrangian Mechanics
This animation shows a disk rolling inside a ring, with the ring resting on two rollers. The friction is assumed to be zero.

Newton’s laws of motion apply here. Given the dimensions and mass of each piece of the apparatus, we could, in principle, use Newton’s three laws to work out how each part of the machine would move over time, and then show the solution in an animation like this. In practice, however, it would be extremely difficult to do this. The Newtonian approach leads into a thicket of interrelated equations that are very hard to solve.

This is where Lagrangian Mechanics comes in. In 1788, French mathematician Joseph Lagrange thought of a way to rewrite Newton’s laws in terms of energy, instead of force. When Lagrange’s method is applied to the rolling disk problem, it is far easier to set up and solve.

Lagrange’s conception was a deep insight into how nature operates, and almost every area of physics, including quantum mechanics, is affected by it.

#physics
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Energy Density
Samsung has recently had to recall all of their new Note 7 phones because there is some danger that its lithium ion battery may catch fire or explode.

Lithium ion batteries are potentially dangerous because they can store a lot of energy (which is why they are used in portable devices).

I thought it would be interesting to compare the energy density of Li ion batteries with some other things I could think of. Energy density is just energy stored, in joules, for each kilogram of mass. The result is the attached table. Note that your phone battery has about twice the energy as a supersonic rifle bullet with the same weight.

Perhaps you can think of some other things that could be on the list.

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James Bradley and the Aberration of Light
In the late 1600’s astronomers discovered a puzzling phenomenon – the position of stars in the sky seemed to shift slightly over time, but were always back at the same position after one year had passed. This shift was called aberration, and its cause was a complete mystery.

In 1727 the English scientist James Bradley thought of an explanation, and the animation illustrates his idea.

As the Earth moves in its orbit from left to right, a photon of light from the star strikes the front lens of the telescope. By the time the photon reaches the eyepiece, the Earth, and telescope, have moved a small distance. Thus, to see the star, the telescope must be pointed slightly in the direction of the Earth’s motion.

Aberration is sometimes compared to running through vertically falling rain – to the person running, the rain seems to be coming at a slant.

The aberration angle depends on the ratio of the Earth’s speed to the speed of light. Knowing the Earth’s speed, and the angle, Bradley computed a fairly accurate value for the speed of light.

#physics #astronomy
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The Catenary Arch
Arches have been used for more than 2,000 years as a way to span a distance without using a beam. However, for most of that time, no one knew what would be the best shape for an arch. All roman arches, for example, were semicircles, though we now know this is not the optimum curve for the arch.

The most efficient shape is the one used, for example, in the Gateway Arch in St. Louis – the catenary curve. In a catenary arch, the force in the arch is always along the curve of the arch. That is, there is no bending force, which would tend to make it buckle and collapse.

The optimum shape was determined by Johann Bernoulli in 1691, using the newly invented calculus. The problem he solved was actually to find the mathematical function for a flexible chain hanging between two points. However, Bernoulli realized that this curve was the same as the one for a perfect arch. With the chain, the only force is tension in the direction of the chain’s curve. In a perfect arch, the only force is compression in the direction of the arch’s curve.

Mathematically, the catenary curve is the function called the hyperbolic cosine.

#physics
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Galileo’s Cannon and Supernova
This animation shows a counter-intuitive phenomenon, but one which can be explained using energy and momentum conservation.

Two balls fall together, the small one being much less massive than the larger one. The large one hits the ground and rebounds. It immediately collides with the smaller one, which is still moving downward. The collision sends the smaller ball upward to a height nine times the height from which the balls were dropped. Energy is still conserved: the small ball’s increase in energy comes from a reduction in the large ball’s energy.

This is sometimes called “Galileo’s Cannon”. I recently used two rubber balls to demonstrate the effect, and it’s a striking thing to see. I drilled a small hole through each ball. A thin rod passes through the holes, and this keeps the balls aligned vertically as they fall.

Two examples that involve similar physics:
- Type II Supernova explosion. When the star runs out of fuel, its core and surrounding gas begins to fall inward. When the core reaches maximum density, it rebounds, then collides with the in-falling gas. The gas is then blown outward at tremendous speed.
- The gravity slingshot maneuver. This is effectively a collision between a spacecraft (the small mass), and a planet (the large mass), though the only contact between the two is via gravity. The spacecraft “steals” some of the planet’s energy, and leaves at a much higher speed that it had before the encounter.

The second image shows a non-mathematical explanation of how this works.

#physics
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Bessel Measures the Distance to a Star
In 1838, German astronomer Friedrich Bessel made the first measurement of the distance to a Star.

The diagram shows how it was done. In the six months that it takes for the Earth to move to the opposite side of the solar system, a star’s position in the sky will shift slightly. The amount of this shift, the parallax, is the small angle at the top of the triangle. The base of the triangle is the width of the solar system. It’s then just a matter of geometry to work out the size of the triangle, and the distance to the star. Obviously, the more distant the star, the smaller the parallax angle will be.

The star Bessel chose was 61 Cygni. The parallax that he measured, using a specially constructed telescope, was only 0.000087 degrees; about the width of a pizza at a distance of 300 km. He then computed the distance to 61 Cygni to be 10.3 light years, which was, at the time, almost unfathomably far away.

Bessel then worked out how bright our Sun would appear to be at that distance. The result was that it would be roughly the same brightness as 61 Cygni, meaning that the stars were almost certainly other suns.

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Pendulum Chaos
A double pendulum is a pendulum with another pendulum attached to its end. It is one of the simplest systems that can have chaotic behavior.

This animation shows the calculated movement of two double pendulums that start from slightly different initial conditions. After a short time, the motion predicted for each of them is completely different. This is the hallmark of a chaotic system – differences in initial conditions, however small, eventually result in large differences in the system behavior.

Note that the mathematical model for the double pendulum is exact, and there is no inherent randomness in the motion. It’s just that we can never know the initial conditions precisely, so that the predicted motion will become less and less certain as we go further out in time.

Two examples of chaotic systems are
1) the Solar System, on a time scale of about 100 million years, and
2) the weather, on a time scale of around 12 days

Animation source: goo.gl/vspShs

#physics #chaos
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Nikola Tesla and His Perfect Electric Motor
In the early days of electric power development in the US, it appeared that a direct current system, backed by Thomas Edison, might win out. One reason was that there were direct current motors, but none for alternating current.

That changed in 1887 when Nicola Tesla invented the motor described here, a type that is still in widespread use. As the first graphic shows, Tesla’s design had three electromagnets spaced 120 degrees apart in the outer, fixed, part of the motor. Each electromagnet is driven by AC current, so that the magnetic field that it produces changes back and forth, varying as a sine wave. At any moment in time, the magnetic field in the center is the (vector) sum of the fields produced by the three electromagnets.

The rotor (not shown) is in the center, where the magnetic fields intersect.

The animation shows how the three vectors add together to produce a net magnetic field (black) that rotates at a constant rate. This rotating field pulls the rotor around, producing power.

It’s a brilliant conception – the rotating field is produced without using moving parts or switches. Also, the strength of the field (i.e., the length of the black vector) is constant, so that the motor’s torque does not vary as the rotor turns, which results in constant, vibration-free, power output.

To show that the three vectors add up in this way, some trigonometry equations are required. A student told me once that trig identities had no practical use, and I said that Tesla’s motor, and our electric power system, might not exist without them.

[added: it’s called a three phase motor, because the input currents fed to each electromagnet are not “in-phase” – their sine wave peaks at different times, 120 degrees apart.]

#physics


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Newton’s Cradle
This is the toy commonly called Newton’s Cradle. Most everyone is familiar with it, and its peculiar behavior- whatever the number of balls that swing down to hit those at rest, is the same number that swing outward on the other side.

Some basic physics is involved in making this happen:

1. _Conservation of Momentum. _ The momentum of each ball is proportional to velocity, and the total momentum of the swinging balls just before a collision is equal to the total momentum of all the balls after the collision.

2. _Conservation of Energy. _ Energy is proportional to velocity squared, and the total energy of the swinging balls just before the collision is equal to the total energy of all the balls after the collision.

With these two laws, the Newton’s Cradle action can be analyzed as a quick series of collisions between pairs of balls, one ball in motion, and the other at rest. Using the conservation laws, it can be show that when a single ball in motion hits a single ball at rest, the balls exchange velocities – the first one stops, and the second moves away at the same velocity.

The graphic shows a sort of movie of the whole collision process, starting with two balls coming from the right (pink) toward two balls at rest (grey). Each frame shows the motion of the balls immediately after a pair of balls has collided. In the last frame, the two on the left are moving, and the two on the right are at rest.

Incidentally, Isaac Newton has no connection to this device – it was invented by a toy company in the 1800’s, and Newton’s Cradle seemed like a catchy name.

#physics #mechanics
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Brownian Motion – Proof that Atoms Exist
In 1827, botanist Robert Brown used a microscope to look at tiny pollen grains on the surface of water. He was surprised to see that the pollen particles were in constant motion, even though they were not alive. This came to be called Brownian Motion, and its cause was a complete mystery.

In 1905, Albert Einstein published a paper that finally explained Brownian Motion. He assumed that the particles moved because they were constantly being hit by unseen water molecules, thousands of times per second. Over a short interval of time, these random impacts would not perfectly balance out, causing the particle to move slightly. Einstein’s theory predicted how far, on average, the particle would drift from the starting point in a given time.

His formulas agree precisely with experimental measurements. This convinced the remaining skeptics that atoms and molecules actually existed.

#physics
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