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Dimples exist on golf balls for a very good reason.  The efect of dimples were actually discovered accidentally, when it was discovered that rough surface, scarred balls traveled further than smooth ones. Ball makers than began to intentionally put dimples on balls to produce the same effect of rough surface balls.

There are two components to drag.  One is frictional drag, but the other is due to pressure drag.  Frictional drag is due to shear stresses on the object.  Pressure drag is due to pressure differences between the front and rear of the object due to boundary layer separation from the surface of the ball.

For streamlined bodies, the drag is mostly due to frictional drag and it is beneficial to have smooth surfaces to promote laminar flow.  However, for blunt bodies like golf balls, drag is mostly due to pressure differences.

For a smooth surface, the boundary layer has less energy and separates from the ball earlier.  The rear just has a wake region that has decreased pressure. A dimpled ball promotes turbulent boundary layer flow.  A turbulent boundary layer has more energy and thus travels further along the surface before separating from the ball.  A dimpled ball thus has less pressure reduction at the rear, and overall less drag.


You can have a burp or a note, but not both.

The Heisenberg uncertainty principle is an interesting non-classical phenomenon, which states that certain pairs of physical properties cannot be known with arbitrary accuracy.  In it’s most common form, it is expressed as an uncertainty between position and momentum, where the position and momentum of a particle cannot both be known simultaneously with arbitrary accuracy.

The idea sounds very foreign to us, as at the macro level, we are familiar with being able to know both the position and momentum of say, a car at the same time.  What is even more interesting, is that the uncertainty between position and momentum is not a limitation on measurements or observation, but actually a description of physical nature itself.  It is a result of quantum mechanics and the wave-like nature of matter.

There is also an uncertainty principle between energy and time.  This uncertainty is commonly known in Fourier analysis as a frequency-time uncertainty.  One can specify a signal with arbitrary time or arbitrary frequency, but not both.  An example of this uncertainty would be the contrast between a burp and a note. For a burp, this signal has a very narrow standard deviation in time-domain (just a single bump), but the signal is comprised of a many different frequencies and has a large uncertainty (standard deviation) in frequency-domain.  The shorter the pulse, the larger the range of frequencies that are used to make up this pulse.  On the other hand, for a flute playing a note, the frequency of that signal has a narrow standard deviation (a single frequency or note), but in the time-domain, it looks like a sine wave and has a large uncertainty (standard deviation).

Similarly, the wave function associated with a particle such as an electron may have a definite position at some time but the momentum will have some uncertainty, or it may have a definite momentum at a given time, but it’s position will be uncertain.

Mathematically speaking, the position-momentum uncertainty principle is expressed as \Delta x \Delta p_x >= \frac{\hbar}{2} where \Delta x is the standard deviation of the position, and \Delta p_x is the standard deviation of the momentum.  While the position and momentum of a particle can be known with arbitrary accuracy, they cannot both be known at the same time.  \hbar = 10^{-34} kg m^2/sec, which is very small.  It is much too small for objects of macroscale size, but is significant for very small objects such as electrons.

Most metals have a crystal structure that is are either body-centered cubic (BCC), face-centered cubic (FCC), or hexagonal close packed (HCP).

The body centered cubic structure consists of one lattice point in the center of the cube in addition to eight corner points.  Many metals such as chromium, iron, tungsten have BCC structure at room temperature.  Visualize it here.

The face centered cubic structure consists of six lattice points at the center of each face of the cube in addition to the eight corner points.  many metals such as aluminum, copper, lead, nickel, and iron crystallize as FCC.  Visualize it here.

The hexagonal close packed structure is similar to the FCC structure, in achieving the highest packing density.  The structures are close packed.    The coordination number or number of closest numbers is maximized (which happens to often minimize the cohesive energy).  Metals such as cadmium, zinc, and titanium have an HCP structure.  Visualize it here.

The simplex hexagonal structure is not close packed.  Visualize the simple hexagonal structure here.  In the simple hexagonal structure, atoms are stacked AAAA.  Atoms are stacked ABAB for hexagonal close packed, and atoms are stacked ABCABC for FCC.

A nanometer is 10^{-9} meters.  To get an idea of how incredibly small a nanometer is, we can take a look at the scale of various things we are familiar with.

At the meter scale, there are humans who are about 1 or 2 meters in height.

One thousand times smaller, at the millimeter scale are fleas or ants or the top of a pin. The diameter of hair is several hundred microns or tenths of a millimeter.

Another thousand times smaller, at the micron scale are cells in our body.

Another thousand times smaller, at the nanometer level, are the diameter of DNA.

An atom diameter is a few Angstroms, or a few tenths of a nanometer.

A good example (borrowed from Richard Feynman‘s Lecture on Physics, Volume 1 Chapter 1 Atoms in Motion) is that if you enlarge an apple to the size of the Earth, then the atoms in that apple are now the size of the original apple.

A nanometer is incredibly small!

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