Ice Skating/ Dancing

Movies of ice skaters 1, 2
 
 
 
 
 

Bicycles & Gyroscopes

Why is it so easy to maintain your balance on a moving bicycle, but not on a stationary one?
 
 
 
 

Kepler's Second Law

This can be restated in terms of the area swept out (dA) in some time interval (dt).
 
 

The area of the triangle can be found by taking 1/2 of the parallelogram's area-

dA= 1/2 R * v dt 

By using the definition of angular momentum (of a point particle) we can rewrite the triangle's area as-

dA= L/ 2m dt

So,

dA/ dt= L/ 2m= constant

Angular momentum has to be conserved as we only have a central force acting on the orbitting body (it cannot exert a torque), so the body sweeps out equal areas in equal time intervals.
 
 

Kepler's second law implies that as a planet's distance from the sun increases in an elliptical orbit, its orbital speed

a. increases
b. decreases
c. remains the same
d. it depends

 
Neutron Stars & Pulsars

When the density of matter gets cosmically extreme, weird stuff happens.  One place where matter can get this compressed is in the interior of dying highly massive stars.  As heavier and heavier elements fuse in the compacted cores of old red supergiants, the core begins to collaspe and eventually falls in on itself, exploding violently in a supernova. Often, the core remaining after the explosion can be so extremely compacted that it is completely made of neutrons.  These insanely dense neutron stars pack the mass of our Sun into a diameter of 30 km, the size of San Diego.  One Dixie cup full of neutron star stuff would weigh around 200 million tons on Earth. 

These stars also have intense magnetic fields which accelerate the electrons which remain in the surrounding gas.  Once the electrons reach a speed near that of light, they must radiate off some of their energy, usually as radio waves or X-rays.  These radio waves are emitted only near the magnetic poles of the star, essentially as two beams in opposite directions.  But,... these stars are rotating at very high rates, so what we see are pulses of the waves, just like a lighthouse's beam that is swept around in circle leaving us to only see pulses of light.