Resonance

A long string, under tension from a weight at one end, is mounted horizontally; the other end connected to a variable frequency mechanical oscillator (right). A transverse standing wave can be set up. This is a well-known class demonstration.

As the frequency is increased, at certain frequencies, a series of large amplitude nodes and loops appears. This is an example of resonance. For more detail, look here

The resonant frequencies are determined by the mass per unit length of the oscillator (bass strings have a natural frequency lower than treble strings), also the tension in the string, thus:

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Resonance occurs when a driven oscillator is made to vibrate at the same frequency as a driver oscillator. Resonance effects are characterised by vibrations with large amplitude.

This link is from a book

It’s well worth a look or two, if only to illustrate the extent of some of these applications in physics and also the fact that simple harmonic  oscillators are nothing more than periodic exchangers of KE and PE.

Here is a link to  Barton’s Pendulums. You will see this in class. In fact, you might make a set yourselves.

Here’s a video produced by a student

Also – see if you can figure out what’s going on here….

Think about a child being pushed on a swing.

The pusher (driver) pushes at the same frequency as the natural frequency of oscillation of the swing (driven). The result is that the swing oscillates with large amplitude, since the energy it needs to gain amplitude is being supplied at exactly the right time in the cycle. The pusher and the pushed are a quarter of a time period out of phase.

These resonance effects are common. The bus side vibrates as it pulls away, since the metal skin of the bus has the same natural frequency as the engine at low revs. Imperfectly balanced washing machines ‘walk’ across the floor during the spin cycle since the frequency of the vertical component is the same as the natural frequency of vibration of the retaining springs, causing vibrations with large amplitude.  Sometimes , these can be disastrous.  Check out the video of the famous collapsing bridge over Tacoma Narrows in Washington State, called “Galloping Gertie”. Amazing! The bridge collapsed in November 1940 because of a resonant effect between the material in its construction and the wind whistling down the gorge, rather like blowing a reed wind instrument to make a loud sound.

Chladni figures are complex resonant vibrations – or two-dimensional standing waves –  made on a flat sheet and this Irish teacher’s take is very good indeed. Originally, Robert Hooke (yes, him again!) experimented with drawing a violin bow across flour-covered glass plates.

Ernst Chladni  (photo) used the same trick with metal sheets, subsequently with a magnetic oscillator in order to fine-tune the frequencies more accurately. They’re quite fun to watch.

Expensive guitar tables have complex Chladni patterns which account for the characteristics of the instrument.

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Chladni figures on a classical guitar

Mass-Spring Oscillator (updated)

Read the IG post on Hooke’s law before proceeding. Check out a Mass Spring Oscillator as a moving .GIF.

In order for mechanical oscillation of a mass/spring system to occur, it must possess two quantities: elasticity and inertia. When the system is displaced from its equilibrium position, elasticity provides a restoring force such that the system tries to return to equilibrium. The inertia property – resistance to changes in motion – causes the system to overshoot equilibrium. This constant interchange between elastic and inertial properties is what allows oscillatory motion to occur. The natural frequency of the oscillation is related to both the elastic and inertia properties. Weak, springy springs (small elastic restoring force) with large weights (large inertia) on them have a long time period and vice-versa.

The weight of the added mass acts vertically downwards, the restoring force on the spring acts upwards.

The elastic constant of the spring is k.

At any other displacement, the acceleration = a

When the mass is displaced downwards the restoring force acting on it, F, is of magnitude kx, upwards. (kx is larger than weight)

If it is displaced upwards the net force (and acceleration) is downwards (kx is smaller than weight)

So:

A graph of T² against m is linear through the origin with slope = 4π²/k

Music of the Spheres

Musical appreciation has much to do with agreeable resonant frequencies.

Pythagoras discovered relationships between musical notes. The pitch of a note being played on, say, a guitar depends on:

  • The length of the string.
  • The tension of the string.
  • The material the string is made of.

Pressing a finger on the string clamps the string on to the metal fret to the left of where the finger is pressed. The effective length of the string becomes the distance between the fret and the bridge. If we say the length of the string is one unit, we can show the effective lengths of the string for each of the notes in the scale as fractions of the whole, related by powers of 2 and 3. These harmonious “consonant” lengths were discovered by Pythagoras. Furthermore, Pythagoreans believed that this was a general principle of the universe: everything is related by the ratios of whole integers..

Soap Chemistry

 

It is hard to say when soap was first invented. Some suppose that even prehistoric man (or, more probably, woman) had a primitive form of it at their disposal. Whether the hunters and gatherers had soap is debatable, but it is certain that soap was available to the ancient Babylonians. Soaps have been excavated in clay cylinders that date back to 2800 BCE. By 1500 BCE. Egyptian medical scrolls recommend a soap made from alkaline salts and animal and vegetable oils for skin conditions.

Later, the ancient Romans discovered the cleaning power of soap accidentally. At Mount Sapo, where animals were sacrificed, rain mixed animal fats, wood ashes and clay into the soil. Incidentally, women washing their clothes by the stream found it was much easier to wash their clothes with some of this clay mixture. Legend links Mount Sapo with the process of soap making (saponification). Interestingly, although Romans are famous for their baths, they actually did not use soap to wash. They coated themselves in oils and then used a scraping tool called a strigil to clean their bodies. However, bars of soap were found in the ruins of Pompeii and archaeologists believe soap was used for laundry and occasionally on the body.

Now for the chemistry. Check out this link. The whole nine yards…

Catalysts

…increase the rate of a reaction while remaining unchanged chemically themselves.

There are two distinct processes…

1. The reactants are ADSORBED on to the catalyst surface, making it easier for the reactants to combine. The reactants like the surface of the catalyst, so they tend to move towards it, making collisions and hence reactions more probable.

2. The catalyst undergoes an intermediate, short-lived reaction, returning to its original state after the reactants have reacted together. Biological catalysts, called ENZYMES work like this.

For more details, download this  Word document explaining how catalysts work

Alcohols

A surprisingly large number of common foodstuffs are produced by direct fermentation of yeast or similar microbes, or by using the principles of fermentation where sugars are digested by micro-organisms to form alcohols.

 

 

 

 

 

 

 

 

 

 

Click alcohols to understand basic alcohol chemistry.

Linear Air Track: Measuring Acceleration (for IG and AS)

The trolley floats on a cushion of air on a V-shaped track, so there’s no friction to slow it down. When the light beam is interrupted by the passage of the trolley it starts a clock. The clock stops when the beam is restored. The logger can be connected to a computer for easier display and configured to calculate average speed.

It's not only me that does all the work around here...
It's not working...Yes, it is.

IG: The card on top of the trolley measured 199mm = the distance travelled. The time for the trolley’s journey through the light gates can be read off the display.

Example:

Average speed = distance / time = 199mm/0.403s = 493.8mm/s (1d.p)

AS students. You MUST read this part: If one end of the trolley was inclined (tipped up), the trolley will accelerate down the slope since the component of its weight acting parallel to the slope will provide the unbalanced force. Can we design and carry out an experiment to compare the acceleration of the trolley with the angle of inclination which will determine the size of the unbalanced force?

I want you to develop your own procedure for this and try it out in the lab, thinking through each stage carefully.

The new light gates are more sensitive and better than the old ones.

If we use two light gates the logger can calculate the average speed of the trolley as it passes through the first and then the second light gate. The onboard timer will then be used to calculate the acceleration of the trolley being the difference in average speed between gate 1 and gate 2.

Perhaps a better way might be to use just one light gate with a split card, like the one shown below.

The lengths of the two uprights and the space between them must be measured. The logger uses (v-u)/t to calculate acceleration.

If we input the dimensions of the card into the logger, it will calculate the average speed during the first and then the second pass of the card and calculate the acceleration for us automatically. If you can’t remember how to do this, look here.

Important design points:

1. How are you going to ensure that the air track is horizontal to start with? Should each run start from the same place? How will you make sure that this happens?

2. You can use plywood spacers or even new exercise books to prop up one end of the track. How are you going to measure the angle to the horizontal? (not with a protractor)

3. What about accuracy? How many times should each measurement be repeated and what are you going to do with the values?  Can you identify any errors in your procedure or results? What are you going to do about them?

4. How are you going to show what your results look like – a table, a graph, or both?

5. Can you predict what the shape of the graph ought to be? What information is obtainable from it?

Solution. Check carefully..All of you should have found time to use the apparatus. AS asks about datalogging principles.

An exaggerated view. The free body force diagram plus resolved components is shown. Weight W acts vertically downwards and normal reaction acts R at 90 degrees to the surface. The resolved component of W, W sin ϑ causes the trolley to accelerate down the slope. F(R) is a zero frictional resistance so can be ignored.