What is energy (mechanical energy)
(Because it cannot be explained, there must be laws. There is no answer, only laws.)
Dynamic, force, potential, energy
equivalent substitution
Money Thoughts
It's easier to see and get the model
Motion
see, touch
Cambrian explosion
(Why does having one more sense make the ecosystem more diverse? Think about the effects on species of the increased dimensions of the ocean, land, and sky in the biosphere.)
Galileo
inertia
big ball small ball
Kepler
Summarized from teacher Tycho’s many years of information
Three major planetary laws (1. Ellipse; 2. Equal area velocity; 3. Period)
Newton
Inverse Square Ratio Orbital, Harley's Cambridge Row, Spawner
Newtonian-Galilean time and space (the flow of time is fixed everywhere, so we can talk about this instant)
What is the speed?
(1) You will have gone far the next time you look at it (Why can’t you just use this? Zino’s Paradox)
(2) v = dx/dt
Based on f = m a
box:?f = m a , a = f / m, where a is a vector (Q: What is a vector?)
What is a?, a = dv / dt (What does the d / dt differential symbol mean?)
sub-box: limit, continuous, differential
What is v?, dx / dt
Supplement: From sums to integrals (bring out Newton to teach us), what better reason is there to review calculus?
∫ f dx ~ Σ?i?f?i?Δx?i
∫ f dx ≡ lim?Δx→0?Σ?N?i=1?f?i?Δx?i?= lim?Δx→0?Σ?N?i=1?f?i?Δx
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mathematical principles of natural philosophy
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Introduction to vector inner product
A?•?B?= |?A?| |?B?| cos θ = AB cos θ
(A?x?, A?y?, A?z?) •(B?x?, B?y?, B?z?) = A?x?B?x?+ A?y?B?y?+ A?z?B?z
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The meaning of vector inner product: (effective direction) effective size related to direction
For example:
1. Effective area of ??capture = fishing net size * projection * interception direction
2. Flow rate = flow rate [inner product] area = flow rate * cross-sectional area
(Why does a small unit area have a direction? Because in the 3-dimensional world, its normal direction is only a specific direction, and the information must be given to be complete.) ()3. The amount of work done = the moving distance * the force received by the object in the direction of movement = the amount of force * the movement of the object in the direction of force applied (bathroom wet and soapy environment)
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The "value" of force
size?
Effect?
Useful or useless?
quantity?
?
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Definition of work
Work W =?f?•?s
(Think: If you just push and don’t move, is there any work done?)
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work-kinetic energy theorem
What does a force do to something with mass (mass is a different degree of inertia)?
If the length and direction of action are not discussed together, the rules will be inconsistent.
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W = ∫?f?· d?x?= m ∫?a?· d?x?= m ∫ (d?v?/dt) · d?x?= m ∫ (d?v?) · (d?x?/dt) = m ∫ d?v?·?v?= m ∫?v?· d?v?= m (1/2) v?2
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conservative force work
Definition of conservative force:
Forces whose work is independent of the process path are called conservative forces
"Path independent" → "Only related to the starting and ending points"
"The process has nothing to do with it" → "Ask only about the results"
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Example of a force that is not conservative: Friction
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How to express the sentence "independent of the path" in mathematics?
W = -∫?f?·?d?r?= U(?r?f?) - U(?r?j?)?≡ ∫dU
(Why negative? Think of the gravity example)
What is dU here? What are its properties?
Calculus: U(x,y,z) is a function of three independent variables
The dU symbol, called the total differential, has the following relationship:
dU = (?U/?x) dx + (?U/?y) dy + (?U/?z) dz
This is a very significant relationship, think about it
ΔU = ? Δx? Δy? Δz?
You can feel it
?
W = ∫ dW?= W - W?0
But originally according to the definition W = ∫?f?·?d?r
Only the above formula ∫?f?cons?·?d?r?= ∫???U?·?d?r?= ∫dU can meet the requirements
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Important conclusion, conservative force must be
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Storage of energy: potential energy (energy related only to position)
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Gravity: f = mg
Gravitational potential energy: U = mgh
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Elasticity: f = - kx
Elastic potential energy: U = (1/2) kx?2
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Supplement: In science and engineering,?f?= -???U is defined to correspond to the force generated by a certain positional energy U.
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Another important conservative force (electrostatic force)
(Why emphasize "static" electric force?)
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From force to beauty: discovering "energy" in force and force action
Force is a vector and energy is a scalar quantity.
Force changes, energy remains constant.
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"Conservation of Energy"
Precisely because energy is transformed through work, it is neither created nor destroyed out of thin air.
Kinetic energy plus potential energy is called mechanical energy (also called mechanical energy) E = K + U
Conservation of mechanical energy: Without the influence of external forces, mechanical energy is conserved
E = K + U
dK = f dr = - dU (don’t leave the good news to outsiders)
Or you can write dU = -fdr = -dK
Therefore dE / dt = d(K+U) / dt = 0 (because dK = -dU => dK + dU = 0 in the above equation)
dE / dt = 0 => E = C (C is a constant, independent of time)
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何謂能量（力學能）
（因為無法說明，所以要有定律。沒有答案，只好定律。）
動、力、位勢、能量
等價的替代
貨幣的思考
看得見，模得到的比較容易
運動
看見、摸著
寒武紀大爆發
（為什麼多一種感官，會讓生態系多樣化增加？想想生物圈海洋、陸地、天空的維度增加，對物種帶來的效應。）
伽利略
慣性
大球小球
克卜勒
從老師第谷多年的資料歸納
三大行星定律（一、橢圓；二、等面積速率；三、週期）
牛頓
平方反比率的軌道，哈雷的劍橋行，催生者
牛頓－伽利略時空間（處處固定的時間流，所以可以談這個瞬間的）
速度是什麼？
(1) 下次看時已走遠 （為什麼不能用這個就好？季諾悖論）
(2) v = dx/dt
基於 f = m a
box : f = m a, a = f / m, 其中 a 是向量（問：向量是什麼？）
a 是什麼？ a = dv / dt （d / dt 微分符號是什麼意思？）
sub-box : 極限、連續、微分
v 是什麼？dx / dt
補充：從 加總 到 積分（請出牛頓來教我們）要複習微積分，有什麼比這個更好的理由？
∫ f dx ~ Σi fi Δxi
∫ f dx ≡ limΔx→0 ΣNi=1 fi Δxi = limΔx→0 ΣNi=1 fi Δx
自然哲學的數學原理
向量內積介紹
A•B = | A | | B | cos θ = A B cos θ
(Ax, Ay, Az) •(Bx, By, Bz) = AxBx + AyBy + AzBz
向量內積的意義：（有效方向）與方向有關的有效大小
例如：
1. 補獲有效面積 = 漁網大小 *投影* 截流方向
2. 流通量 = 流速〔內積〕面積 = 流速 * 截面積
（小單元面積為什麼有方向？因為在 3維 的世界裏，其法方向只是某特定方向，必須給定訊息才完整。）（）3. 作功大小 = 移動距離 * 物體在移動方向上的收力 = 施力大小 * 物體在施力方向上的移動（浴室溼滑肥皂境）
力的「價值」
大、小？
效、果？
有功、無益？
數、量？
功的定義
功 W = f•s
(Think : 只推，不動，到底有沒有作功？)
功－動能定理
力對帶有質量的東西（質量是不同慣性的程度），會做什麼？
不一併探討作用長度及方向的話，就會漫無法則。
W = ∫ f · dx = m ∫ a · dx = m ∫ (dv/dt) · dx = m ∫ (dv) · (dx/dt) = m ∫ dv · v = m ∫ v · dv = m (1/2) v2
保守力作功
保守力的定義：
作功與過程路徑無關的力，稱為保守力
「路徑無關」→「只與起始與終結點有關」
「過程無關」→「只問結果」
不是保守力的例子：摩擦力
「與路徑無關」這句話，怎麼用數學表達？
W = -∫ f · dr = U(rf) - U(rj) ≡ ∫dU
（為什麼是負的？想想重力的例子）
這裏的 dU 是什麼？它有什麼性質？
微積分： U(x,y,z) 是三個自變數的函數
叫作全微分的 dU 符號，有以下關係：
dU = (∂U/∂x) dx + (∂U/∂y) dy + (∂U/∂z) dz
這是意義重大關係式，想想
ΔU = ？ Δx ？Δy ？Δz？
就可以體會
W = ∫ dW = W - W0
但本來依定義 W = ∫ f · dr
唯有上式 ∫ fcons · dr = ∫∇U · dr = ∫dU ，才能符合要求
重要結論，保守力必定
能量的儲存：位能（只跟位置有關的能量）
重力 : f = m g
重力位能：U = m g h
彈力：f = - k x
彈力位能：U = (1/2) k x2
補充：在科學及工程上，定義 f = -∇U 來對應某持定 位能 U 所產生的力。
另一個重要的保守力 （靜電力）
（為什麼強調 "靜"電作用力？）
從力到美：在作用力、力作用中發現「能量」
力是向量、能量是純量。
力是變化的，能量保持定值。
「能量守恆」
正因為能量是透過作功來變換，因此不頓外產生，亦不憑空消滅。
動能加位能，叫作力學能（又稱機械能）E = K + U
力學能守恆：無外力的影響下，力學能守恆
E = K + U
dK = f dr = - dU （肥水不落外人田）
或是寫醬也可以 dU = -fdr = -dK
因此 dE / dt = d(K+U) / dt = 0 （因為上式面的 dK = -dU => dK + dU = 0）
dE / dt = 0 => E = C （C 是某常數，與時間無關）