p + ρgz = constant  [Bernoulli's equation]

p, ρ都是z的function, 但differentiate兩邊against z

dp/dz  + ρg + dρ/dz  gz  = 0
dp/dz   =  – ρg – dρ/dz  gz
這是不正確，沒有physical meaning的，正確的應是:
dp/dz = – ρg

solution:
=> energy modes for these system: rotational kinetic and torsional potential energy
=> equipartition theorem applies to both types of disordered energy
=> 1/2 K θ_rms ² =1/2 k T
=>  T = 290K

• set zero / 計offset
• 較intensity / attenuation , unit
• distance, polarization, background, 有無focus好, d野係咩parallel, 做左calibration未

http://en.wikipedia.org/wiki/Function_generator

Signals which last for a very short period of time are referred to as “ TRANSIENT Signals”.

http://en.wikipedia.org/wiki/Coupling_(electronics)
In electronics and telecommunication, coupling is the desirable or undesirable transfer of energy from one medium, such as a metallic wire or an optical fiber, to another medium, including fortuitous transfer.

Also the transfer of power from one circuit segment to another. E.g., an alternating voltage may be transferred to a segment at a different direct voltage by use of a capacitor or transformer; power may be efficiently transferred to a segment with different impedance by use of a transformer.

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larger degree of freedom => larger energy => entropy (in positive temperature) is larger
越自由越凌亂
[物理看人生 = ="]

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點解mixture of liquid的boiling point低d
=>http://www.jyu.fi/kastdk/olympiads/1998/ps.zip （睇第一題）

neutron [matter-wave] interference, using gravity
=>http://www.jyu.fi/kastdk/olympiads/2006/IPhO-37_Th1_final.pdf

What Does Negative Temperature Mean?
-300K 「熱過」 273K

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物理一題（F.3程度的熱力學, F.4 Chem的知識）：
The amount of heat required to convert 1g of ice into vapour is approximately 3000J. Estimate the energy required to separate one molecule from a single neighbour, assuming that, in ice, each H2O molecule has four nearest neighbours. [Relative molecular mass of water=18, Avogadro constant=6.02 x 10 ²³ mol ^-1]

4.5 x 10^-20J
特別之處解法有點數學技巧。
Graph theory: The sum of the vertex degrees in any undirected graph is twice the number of edges.
簡單來講，就是如果在一個宴會上，握手次數共為m次，且每兩個人之間最多握一次手；則 甲握手次數+乙握手次數+… =2m
From “Cambridge Problems in Physics and Advice on Solutions”

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http://terrytao.wordpress.com/2007/12/28/einsteins-derivation-of-emc2/
Einstein’s derivation of E=mc^2

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http://youtube.com/watch?v=F5PvIPgJGx0
Kelvin water dropper

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Curley Hair Gets Less Tangled Than Straight Hair.

The hair on people’s heads (typically 100,000-150,000 hairs per head) comes in lots of shades, degrees of oiliness, and amounts of curliness. Jean-Baptiste Masson, who works at the Laboratory for Optics and Biosciences of the Ecole Polytechnique in France set out to study the problem scientifically. On the experimental front, he consulted hairdressers and got them to count tangles in people’s hair. On the theoretical front, he devised a geometrical model of hair, hoping to explain the results mathematically.

Tangles, defined as groupings of hair that resist combing, proved to be almost twice as prevalent with straight hair than with curly hair. Masson (jean-baptiste.masson@polytechnique.fr) explains this by saying that although straight hairs interact with each other less frequently the interaction is at great angles, and it is the relative angle between hairs that causes tangles. This in turn is a consequence of the surface properties of the hairs. One possible application of this work on hair, Masson says, is in designing velcro-like products.

Q1. A binary star system consists of two revolving round a common centre with different orbital radii. If the period of  rotation is 10days and their masses are 10^30kg and 4×10^30kg respectively, find the seperation between them.(Circular motion and G-force)

Q2. The number of electrons in your body is roughly:
A. 10^20 B. 10^23 C. 10^27 D. 10^30

Ans:

Q1:  1.85×10^10m  (source: forgotten)

Q2: C (source)

PhO training
下午tutorial，tutor講了三小時talk。好正呀。

• 怎樣的physical quantities才算有physical meaning? 可以用實驗「直接」度量出來。
• 有時，為了計算上的方便，物理學家定義一些quantities出來，後來才證實它有physical meaning。
• 物理和數學如何互相影響。所謂的physical approach和mathematical approach，前者可說成geometical approach (symmetry)，後者可能關係更多algebra。關係有點似數學比賽中用coordinate geometry老屈pure geo的問題。
  • positive mass conjecture：廣義相對論說，兩個masses的interaction得出的”mass”，會被兩個mass直接加少。我們直觀地相信全宇宙的物質的total mass是正數，但General Relativity似乎說有可能是負！於是人們便去證明。丘成桐成功用數學證明了，但後來Witten卻從物理角度，給出一個簡單且可推廣的證法。
• string theoists 提出了問題數學家做（註：個人感想：等某些數學家不用乞食 =.=”）。
• 怎樣比較哪樣quantities更fundermantal?（比較 force和momentum, energy，electric field和potential，magnetic field和vector potential，雖然人類都是先認知到前者的現在，後者卻更重要。）
• Gauss’s law in differential form和Coulomb’s law ，兩條公式可以互相推導出，為何前者更「fundermantal」？因為後者的引伸意義只是inverse square law，前者從微分幾何的角度，可以說明E和r^{-1+n}（n是空間維度），亦可以引伸不同的對稱性。

• 	slow	fast (compare with light speed)
  macro	Newtonian	special relativity
  micro	quantum mechanics	quantum field theory

• 企圖統一quantum field theory和general relativity的string theory, string nets, quantum hoop… 等等的理論，是在建立theory of everything（解釋到世上一切的事物），還是theory of anything（不論存在在我們的自然與否，都強大得可以解釋）?
• 點解string theory畀人話（註：我聽過一個說法，是string theory主要不是「解釋」現象，更多「製造」現象，不過他說的是另一個論點）。實驗證明不到，已經不是science。其實做科學好現實，商業講要有錢，科學講求要有實驗去判斷真假。

「這個principle並不理所當然，在四種基本力中，萬有引力和電磁力都符合這個principle，但強核力和弱核力並不符。……例如，如果老師在旁，你對你的同學的態度，是否跟老師不在時一樣呢？」

今日講一個CE Physics的嚴重錯誤（這是F.3就接觸到的）。
如何定義溫度（temperature）？
如果說是用於「冷」和「暖」的比較，因為「冷」、「暖」是主觀的，溫度是客觀的。比較冷暖，是溫度的功用，而非定義。
F.3的物理書亦有提及是「measure of average kinetic energy of particles」，這種說法，其實只適用於理想氣體（ideal gas），或者特定物質在特定態和特定壓力之下。

為甚麼不成立呢？
考 慮一塊在雪櫃內的冰。冰和雪櫃空氣溫度相同。然而，冰中的粒子，和雪櫃中空氣水蒸氣的粒子，化學結構同為H2O，因此，計算平均動能KE = 1/2 m v^2 的 m 是相同的。然而冰中的粒子的平均速率，顯然大大低於水蒸氣粒子的平均速率，因此平均動能也有很大差距。
“There is no one-to-one relationship between energy content and temperature; they are quite distinct concepts.” – Statistical and Thermal Physics, M. D. Sturge

那 樣我們可以怎樣定義「溫度」呢？且慢！「溫度」是否存在呢？不，我們連溫度都未知道是甚麼，怎知它存在與否。結果，先要想想我們當初為何要追求「溫度」。 在熱力學，溫度是隨着zeroth law of thermodynamics而產生的。故此，正確的問法，是「能否定義出一個標準（溫度），使得：不同系統在thermal equilibrium，若且唯若，系統的某特徵（溫度的大小），是相同的」。
上面提到的書中有一個證明，證明zeroth law引伸了溫度的存在。當中的重要假設是有兩個

1. 「three objects, labeled by i=1,2,3, as fixed quantities of a compressible fluid, each of whose states can be defined by measuring the pressure p_i and volume V_i.
2. 「… equilibrium is established, it is found that p and V can no longer be varied arbitarily」。
問題是，這兩個結果是如何「found」的？
如果是由理論求得，我們又可以追問理論上的假設是否成立，然後落入更深更大的問題（如what is pressure => what is force?）。
如果是由實驗求得，而這個推論成立，我們用一支水銀溫度計便足以指出「溫度」存在。（這也是我相信足夠解釋的一切。