028 Can you blow your own sail?

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Basically, the air gave the cart a backward force when they are pushed forward by the fan, and a forward force when they collide into the sail.  However, since these two forces need not have the same magnitude (they are not action-reaction pair), it is unclear which direction the net force will be.

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The principle of conservation of momentum can make the analysis easier. Just use the fact that the total momentum of the cart plus the air as a system must remain as zero!

With the original sail, the air got pushed out around the sides of the sail. Since the net change in momentum of the air is zero, the cart cannot have any change in momentum.

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With the improvised sail, the air got turned around and back. Since the air have acquired a backward momentum, the cart must acquire a forward momentum.

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In practice of course it will be silly to blow your own sail forward. It is much easier (and more efficient) to simply blow backward.

027 Faraday’s Law

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When the magnet was moving slowly, the LEDs did not light up because the magnetic flux linkage was changing too slowly. There was an induced emf, but it was too small (Faraday’s Law: ε = -dΦ/dt) to light up the LEDs. (About 1.5V is required to forward bias the LEDs)

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The LEDs also did not light up when the long magnet was moving vigorously in the coil. This is because since the entire magnetic flux of the magnet was already captured by the coil. So even though the magnet was moving rapidly, the magnetic flux linkage remains constant. Hence no emf nor current was induced in the coil. (Faraday’s Law: ε = -dΦ/dt)

026 Faraday’s Law

First of all, you probably figured out that the two LEDs were both connected in parallel, directly to the coil, but in opposite directions. This would explain why they do not light up together. (LEDs are diodes, so they allow current to flow in only one direction).

Let’s move on to the EMI part of the explanation.

Firstly, Faraday’s Law says that if the magnetic flux linkage of a coil Φ changes, an emf ε directly proportional to the rate of change of Φ will be induced in the coil. (Faraday’s Law: ε = -dΦ/dt)

When the magnet was pushed into the coil, Φ increased, thus inducing an emf and current in one direction in the coil, lighting up the red LED. When the magnet was pulled out of the coil, Φ decreased, thus inducing an emf and current in the coil in the other direction, thus lighting up the green LED.

To analyze such situations, it is usually useful to sketch the variation of Φ with time. We can then tell the induced emf ε from the gradient of the Φ-t graph. (Faraday’s Law: ε = -dΦ/dt)

Hopefully you will find the following diagrams self-explanatory.

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Magnet pushed in and pulled out of coil (North pole facing downward)

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Magnet pushed in and pulled out of coil (South pole facing downward)

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 Magnet dropped through the coil (North pole facing downward)

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025 AC lighting

First of all, the street lamps are pulsating because they are powered by an alternating current. Singapore uses 50 Hz AC supply, so the street lamps pulsate 100 times per second.

Some people expect 50 pulsations per second. They have forgotten that a lamp lights up two times in each AC cycle: once during the positive half-cycle, and once more during the negative half-cycle.

024 Why does this DC motor not need any commutator?

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When electric current passes through the coil in a magnetic field, the magnetic forces (F=BIL) produce an anti-clockwise torque which turns the DC motor.

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When the coil crosses over to the other side, the magnetic forces produce a clockwise torque instead which would slow down the rotation. This is of course undesirable.

Usually, a commutator is used to reverse the current (and thus magnetic forces on either sides of the coil) during each cross over to keep the torque in the same direction.

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What this video showed is a cheap alternative to the commutator: The coil’s insulation was sandpapered away only on one side. So when the coil flips over, it stops making electrical contact with the power supply. Without any current in the coil, there is no magnetic force. So the coil only experiences an anti-clockwise torque for half a cycle, and zero torque for the other half. Now we have a commutator-less DC motor that works by working by working only half the time.

Note: To be accurate, the connection is switched off completely only when the coil is horizontal. So the clockwise torque is not completely zero, but it is decreased enough for the design to work.

023 Why did the 35 W bulb outshine the 50 W bulb?

A bulb with a 12 V 50 W rating is designed to have a resistance of 12²/50 = 2.88 Ω when in operation (P=V²/R). Similarly, a bulb rated at 12 V 35 W rating is designed to have an operating resistance of 12²/35 = 4.11 Ω. The point to note is that a 50 W bulb has a smaller resistance than a 35 W bulb.

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At first both bulbs were individually connected across a 6 V battery. Since the potential differences across both bulbs were the same, to compare the power dissipated in the two bulbs, we should think V²/R. So the 50 W bulb, with a smaller resistance, shone brighter.

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Later the bulbs were connected in series across a 12 V battery. Since the current flowing through both bulbs was the same, to compare the power dissipated in the two bulbs, we should think I²R.  So the 35 W bulb, with larger resistance, shone brighter.

(Do note that when connected in series, the potential difference across each bulb was no longer 6 V each. By the potential divider principle, the 35 W bulb, with its larger resistance, ended up with more than 6 V of potential difference across, while the 50 W bulb had less than 6 V across)

022 Tuning Fork Interference

From the video, it was clear the loudness as picked up by the microphone changes with the orientation of the tuning fork. In fact, when the tuning fork is turned diagonally away from the microphone, there was complete silence. The cause of the dips in loudness was destructive interference.

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Most of us don’t realize that the tuning fork is not a single sound source. Whenever the outer surfaces are pushing outward and propagating compressions, the inner surfaces are propagating rarefactions. And whenever the outer surfaces are retracting inward and propagating rarefactions, the inner surfaces are propagating compressions.

So we should actually model the tuning fork as 4 coherent sound sources, one pair of in-phase A sources (formed by the outer surfaces) which are completely out-of-phase with the other pair of B sources (formed by the inner surfaces).

The interference of these four sound sources causes the loudness to vary with direction in the manner depicted by the directivity pattern shown below.

Do note that the dimension of the tuning fork is small compared to the wavelength, so regardless of orientations, sources A are always are always interfering destructively with sources B. However, destructive interference is least complete at 0° and 90° because of the difference in intensities between the source A waves and source B waves (remember the inverse square law). Complete destructive interference occurs only at 54°, which is why there is complete silence at this angle.

For more detail, please refer to http://www.acs.psu.edu/drussell/publications/tuningfork.pdf

021 Tuning Fork Resonance

This is a demonstration of the phenomenon called resonance: the amplitude of a forced oscillation is largest when the driving frequency matches the resonant frequency.

 

In this video, the sound wave emitted by one tuning fork is forcing the other tuning fork into oscillations. At first, the frequency of the sound wave matches closely the resonant frequency of the tuning fork. This allows for energy to be transferred from the sound wave to the tuning fork efficiently. The forced oscillation thus reaches a large amplitude, leading to an audible hum.

The driving tuning fork was then clamped to change its pitch. The sound wave it emits no longer matches the resonant frequency of the other tuning fork. As such, energy is transferred from the sound wave to the driven tuning fork less efficiently. The amplitude of the forced oscillation was so small nothing was picked up by the microphone.