Somebody introduced this demo to me as a diffraction phenomena. But it can’t be. Because visible light, with wavelength less than 1 um, cannot possibly diffract so much around a card. Anyway, I tried using red and blue light (instead of white), but did not notice any difference. If it were diffraction, changing the wavelength must have an effect.
I believe lengthening of the nose can be explained simply by the darkening of the penumbra of the nose into umbra. When the penumbra of the card meets the umbra of the nose, the card began to cut off the light that would have fallen on the penumbra of the nose. I guess we can think of the the card as making the extended light source more and more like a point source. The penumbra of the nose near the umbra progressively becomes the umbra, thus extending the umbra of the nose outward.
This demo also made me realize that the penumbra is not a region of continous intensity (unlike how it is often drawn in textbooks). There must be a gradient in intensity from completely dark to completely bright across the the penumbra. I also realized that the penumbra is wider than what our brains interpret it to be.
First, take note that while the pendulum is capable of oscillating in two directions, the pendulum lengths in the two directions are slightly different.
If the periods of oscillation for both modes were exactly the same, the pendulum would be drawing the same straight line in the direction diagonal to the two modes of oscillation, which would be kind of boring. But because the periods of oscillation in the two directions are slightly different, oscillation in one direction always completes just a little earlier than oscillation in the other direction. As result, the pendulum ends up drawing ellipses that spin in then clockwise direction, then anti-clockwise, and so on.
What makes the pattern even more interesting is the fact the osccilations do decay as time progresses due to damping. As a result, the ellipses shrink towards the centre as time progresses.
See also the pendulum in fast motion.
If the solenoid were powered by a DC current, the ring should only experience a changing magnetic flux only at the instant when the solenoid was turned on. The ring would then have jumped only once, before falling back to the bottom. The fact that it is able to hover tells us that it continuously experiences an upward magnetic force that balances its downward weight.
This is possible only if the solenoid is powered by an AC current. We can in fact thinking of the solenoid as the primary coil of a transfomer, and the ring as the secondary coil. The primary and secondary currents are always in-phase (or completely out-of-phase if you choose). So the current in the solenoid and the ring produce magnetic fluxes which are always opposing each other, resulting in a continuous upward magnetic force (a sine-square varying force, to be exact) on the ring, which keeps the ring in the air.
Let’s consider the system of the bike and the rider as one rigid body.
Since the system is undergoing circular motion, the resultant force acting on it must be in the centripetal direction with magnitude mv2/r. So we can write down the two equations:
(vertically): N = mg
(horizontally): f = mv2/r
Next, since the system is not rotating (it maintains the lean angle θ), the resultant moment acting about the centre of mass of the system must be zero*. This implies that the contact force (the resultant of N and f) must be directed towards the centre of mass.
(taking moments about the C.M.): f/N = tanθ
Putting the three equations together, we get
v2/rg = tanθ
So centripetal acceleration v2/r = gtanθ. So if we see a bike leaning at 45°, the biker is experiencing about 1g of acceleration in the centripetal direction. It will be 2g at 63°, and 3g at 72°.
*This is a simplified analysis. People familiar with precession will know more. However, for A-level purpose, this simplified analysis is more than sufficient.