Month: June 2014

044 Circular Motion of Cars and Motorcycles

http://www.youtube.com/watch?v=eiDUlqI_cK4&rel=0

For cars to turn, the wheels must be turned into the bend so that the frictional force between the road and the tyre has a component perpendicular to the current direction of travel of the car. This component of friction provides the required centripetal force for circular motion.

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This centripetal force, unfortunately, has a “side effect”. It produces a torque about the center of mass of the car which tilts the car away from the bend. Why does the car not flip over? Well, if the car turns too sharply, it does flip over. But most of the time, as the car leans outward, the contact force on the inner wheel (which tends to flip the car) drops, and the contact force on the outer wheel (which opposes the flip) increases. This restores rotational equilibrium to the car so it does not slant any further.

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This “self-correction” mechanism is however not available to motorbikes since they have only one set of wheels. Instead, motorbike riders must learn to lean into the bend. The shifting of the centre of mass allows them to achieve rotational equilibrium.

043 Barton’s Pendulum (Phase Relationship)

Do notice that the pendulums have different phase relationships with the driver.

1) The two shorter pendulums oscillate roughly in-phase with the driver.

2) The two longer pendulums oscillate roughly completely out-of-phase with the driver.

3) The pendulum in resonance lags the driver by about a quarter of a cycle (or PI/2 radians).

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This kind of makes sense. Because if the driver’s displacement leads the pendulum’s displacement by a quarter of a cycle, it actually means that the driver’s displacement is in-sync with the pendulum’s velocity. This means the driver is always tugging the pendulum in the same direction as the pendulum’s velocity1. This means the driver is in the position to be doing positive work and inputting energy to the pendulum all the time. The pendulum goes into resonance because the transfer of energy from the driver to the pendulum is at its most efficient.

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On the other hand, if the driver’s displacement is in-sync with the pendulum’s displacement, it actually means that the driver’s displacement is one quarter cycle behind pendulum’s velocity. This means the driver is tugging the pendulum in the same direction as the pendulum’s velocity only half the time. The other half of the time, the driver is actually tugging the pendulum in the opposite direction as the pendullum’s velocity, slowing the pendulum down. The net work done by the driver on the pendulum is close to zero. No wonder the pendulum oscillate at small amplitudes.

The same thing happens if the driver is completely out of phase with the pendulum.

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It can be shown using complicated mathematics that the phase relationship between the driver and the forced oscillation depends on the mismatch between the driving frequency and the resonant frequency. This goes in some way to explain the shape of the resonance curve.

1. Note that whenever the driver is displaced to the right, it is tugging the pendulums to the right. Likewise, whenever the driver is displaced to the left, it is tugging the pendulums to the left. In other words, the the periodic driving force is in-sync with the displacement of the driver.

042 Barton’s Pendulum

Each of the five pendulums have a different resonant frequency due to their different lengths. The one whose resonant frequency matches the driver’s frequency oscillates at maximum amplitude (resonance).

Do note that whether resonance or not, all the pendulums were forced to oscillate at the driver’s frequency (and not their natural frequency).

Practical notes:

  • The oscillators must be damped if not it will take too long for the natural oscillations to decay.
  • Actually, better results are obtained if the order of the pendulums is swapped.

 

041 Perfectly Inelastic Collision

The Principle of Conservation of Energy and the Principle of Conservation of Momentum are universal laws applicable to ALL situations, including collisions. So the total energy and total momentum  are always conserved in ALL collisions.

There is however nothing which dictates that Kinetic Energy must be conserved.

– In an (perfectly) elastic collision, all the KE is conserved.

– In an inelastic collision, some KE is “lost”1.

– A perfectly inelastic collision is the one that loses the most KE2.

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A “classic” collision is that of a body colliding into another stationary body of equal mass. If the collision is perfectly inelastic, the outcome is that of both bodies traveling at half the speed after the collision. Note that this outcome results in the maximum loss of total kinetic energy AND conserves momentum.

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Another “classic” collision is that of two equal masses travelling at the same speed colliding head on into each other. If the collision is perfectly inelastic, the outcome is that of both bodies coming to rest after the collision. Again, note that this outcome results in the maximum loss of total kinetic energy AND conserves momentum.

In fact, it can be shown mathematically that for a two-body head-on collision, having both bodies travelling at the same speed after the collision is the outcome that maximises the loss of kinetic energy. So having two bodies “stuck together” after the collision is the tell-tale sign for a perfectly inelastic collision.


1. The kinetic energy of the colliding bodies could have been passed on to (1) vibrational energy, manifesting as heat and sound, or (2) potential energy in the molecular bonds, manifesting as permanent deformations in the colliding bodies.

2. It is not possible to always possible to lose ALL kinetic energy in a collision because of the Principle of Conservation of Momentum. If the total momentum before the collision was not zero to start out with, the total momentum after the collision cannot be zero either.

040 Damping and Resonance Amplitude

A forced oscillation attains its final amplitude when the rate of input of energy (from the driver) matches the rate of loss of energy (to the surrounding).

The rate of loss of energy is dependent on (1) the amplitude of oscillation and (2) the degree of damping. As the amplitude of the forced oscillation grows, the rate of energy loss due to damping also increases.

Picture2So if the degree of damping is high, the forced oscillation will reach the equilibrium state at a smaller amplitude. (Kind of similar to why larger air resistance results in lower terminal velocity for falling objects) This explains why the stronger the damping, the lower the resonance peak.

^ In theory, if there is no damping, the amplitude of forced oscillation should grow indefinitely. In practice, however, the oscillation would have broken down at some stage to end the party.

039 Resonance

There are two ways to set a pendulum into oscillation.

<00:07> Free Oscillation

One way is to displace it from its equilibrium position, and then release it. We get what is called a free oscillation, instantly.

A second way is to continuously feed energy into the pendulum through a periodic driving force (provided by a driver). This is called a forced oscillation.

<00:21> Driver at High Frequency

With the driver oscillating at a very high frequency, the pendulum is driven into oscillating at the same very high frequency. But at this high frequency, the pendulum can barely react to the driver, so the amplitude of the forced oscillation is very small. In fact, if the frequency of the driver were to become infinitely high, the pendulum will practically stay motionless.

<00:48> Driver at Low Frequency

With the driver oscillating at a very low frequency, the pendulum is driven into oscillating at the same very low frequency. But at this low frequency, the pendulum is merely moving in line with the driver, resulting in a small amplitude. In fact, the amplitude of the pendulum is exactly equal to the amplitude of the driver.

<01:12> Driver at Natural Frequency

With the driver oscillating at the natural frequency of the pendulum, the pendulum is driven into resonance. By going along with the pendulum’s natural tendency, the driver has found the most efficient frequency at which to continuously transfer energy to the driven. As result, the amplitude of the pendulum grows and grows with each stroke of the driver.

The phenomenon of resonance is nicely presented in resonance graphs, which plots the amplitude of the forced oscillation against the frequency of the driver.

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At very low frequency, the amplitude approaches the amplitude of the driver. At very high frequency, the amplitude approaches zero. Something special happens somewhere in the middle. At the resonant frequency, which is practically the natural frequency of the free oscillation, the amplitude of the forced oscillation reaches its peak. This is called resonance.

 

038 Penetrative Power of Nuclear Radiation

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Alpha particles are highly ionising, but have very little penetrative power as they lose their energy rather quickly. All it takes is a few cm of air or a piece of tissue paper to completely bring alpha particles to a stop.

Beta particles are less ionising, but more penetrative compared to alpha. Depending on how energetic the beta particles are, it a few mm of aluminium to soak up the beta particles.

Gamma radiation are really EM radiation. They are only weakly ionising, but have very high penetrative power. It takes a thick piece of lead to stop gamma.

With these facts, it is clear that this radiation consists of predominantly alpha particles. There rest are probably beta particles, and perhaps even gamma radiation.

The source is actually Americium-241 obtained from a smoke detector. Am-241 is an alpha-emitter with a half-life of 432.7 years. It does emit a little bit of gamma radiation. Its daughter nuclide Neptunium-237 is a alpha-emitter with a half-life of 160 000 years. The grand daughter nuclide Protactinum-233 is a beta-emitter with half-life of 27 days.

037 Incandescence, LED and Fluorescent Spectrum

This video showcases the spectrum produced by three different types of lighting.

(1)

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A tungsten filament glows because of electromagnetic radiation generated by the thermal motion (more specifically, acceleration) of charged particles. Production of light in this manner is called thermal radiation or incandescence. Since there is a continuous range of thermal motion, incandescence produces a continuous spectrum.

(2)

300px-PnJunction-LED-E.svg

LED is a p-n junction that emits photons when conduction band electrons recombine with valence band holes. The energy (and thus wavelength) of the photon is equal to the energy transition made by the electron. This is kind of similar to how light is produced in a gas discharge lamp. However, unlike a gas atom where transitions are between discrete energy lines, in the p-n junction the transitions are between the conduction band and valence band. The spectrum of a LED light is thus not discrete, but centred about the band gap energy.

(3)

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A fluorescent tube is like a improved  mercury discharge tube. Unlike a mercury vapour lamp where a lot of energy is wasted in the ultra-violet light produced, a fluorescent tube is coated with a fluorescent coating. The fluorescent molecules are excited by ultra-violet photon from ground state to one of the vibrational states in the excited states. When they de-excite (through collisions with other molecules), they emit a few visible photons as they cascade from the vibrational states. As a result, the spectrum of a fluorescent lamp consist of both the discrete lines due to the mercury vapour and a continuous spectrum due to fluorescence. The choice of fluorescent material gives rise to different flavors such as warm white, cool white, daylight, etc.