1. The previous article introduces that the rotation of the Earth can influence the direction of wind and affect the propagation of polar air masses in a simple setting. Air masses originating from the North Pole will first deflect to the right. To know more about what is Coriolis Force, the physical interpretation of Coriolis effect is of the fundamental importance. Here, we shall start looking at this without any formulaic expression of Coriolis effect. (which can be a headache to those who dislike numbers and symbols)
2. The concept of Coriolis effect adheres to the concept of "reference frames" in physics. Reference frame can be easily understood by the following analogy.
3. Suppose there is a platform in the train station and suppose there is a train moving westward at 100 km/h passing through the platform. Now, consider Thomas is standing on the platform and Henry is standing on the train.
4. From the perspective of Thomas, Thomas himself is standing still and Henry is moving westward at 100 km/h, which is in the same direction and at the same speed of the train.
5. From the perspective of Henry, Henry himself is standing still on the train. Thomas is moving eastward at 100 km/h. In fact, the platform is also moving eastward at 100 km/h.
6. Therefore, the perspective of Thomas and perspective of Henry form distinctive frames of reference. The reference frame of Thomas is to "view" everything with the reference to the platform. The reference frame of Henry is to "view" everything with the reference to the train. (i.e. the platform and the train are two separate reference frames.) It is correct to say either one of them is moving or at rest, depending on which reference frames you are referring to.
7. The reference frame which possesses the following characteristics is called Inertial Reference Frame.
- No change in velocity (i.e. no acceleration)
- Moving in a straight line
So, the analogy above demonstrates the characteristics of Initial Reference Frames. By definition, other reference frames without one of the above 2 characteristics are called Non-inertial Reference Frame.
8. The adjective "inertial" comes the word "inertia".
9. The Newton's first law has defined Inertia in this sense:
"Without exerting additional force, the object, whether it is at rest or moving forward in a straight line, is resisting to the change of its velocity."
10. This is the reason why a vehicle will not suddenly move since the mass of the object prevents any change of velocity. On the other hand, objects which is at first moving forward will not decelerate suddenly without external forces (which implies a change in velocity). At the time when Newton comes up with this idea, it is difficult to accept because at the first glance, objects on the Earth can stop spontaneously. For example, balls will stop when they traveled distant enough. But, in fact, deceleration of objects are due to friction on the surface of ground. Newton's Laws of Motion are always interpreted separately with fiction - deceleration will not occur unless an addition force has affected the movement of object.
11. In the inertial reference frame, the Newton's laws can be easily fitted without any adjustment for the purpose of describing the motion of the objects. Every motion and activity can be described by the Cartesian Coordinate System. Here are several simple examples:
From Henry's perspective,
Thomas is travelling in opposite direction.
A boat is surfing slowly.
Another train is standing still.
A balloon came closer to Henry.
Without additional force, all of them are moving in the same speed at the same direction forever.
12. The Earth is a Rotating Reference Frame (a type of non-inertial reference frame) as a result of the Earth's rotation at an angle of 23 1/2 degrees.
13. When the Newton's laws of motion applies to the area of non-inertial reference frame (such as rotating reference frame), fictitious force (also known as Inertial Force) must be incorporated so that the laws will give a valid prediction and explanation on the motion of objects, creating the same prediction of motion as they are used to predicts motion of objects in inertial reference frame.
14. Therefore, the fictitious forces are something extra for the ease of describing motions just as what is done in inertial reference frame. These forces also render the Newton's Laws of Motion to be useful in describing things in Rotating Reference Frame, without being distorted by the circular motion and background rotation in a rotating reference frame.
15. Suppose Henry pushed a ball from the origin of the circle towards the circumference on a large roundabout. Suppose Thomas is sitting on the edge of the roundabout and would like to take the ball. Suppose there is no rotation on the roundabout. Now, Henry throws the ball out from the centre of the roundabout towards Thomas. Without any additional force, the ball follows a straight line motion due to inertia. From the perspective of Henry and Thomas, they are all agreed on the straight line motion. Therefore, no modification by adding fictitious force is needed to describe the motion of the ball.
16. Now, suppose someone has turned the roundabout anticlockwise barely. Henry will NOT move since he is sitting on the origin of the roundabout. However, Thomas who is sitting on the circumference, started to rotate. Let's assume that in 1 minute, the roundabout turned anti-clockwise by 45 degrees.
17. From the perspective of Henry, he finds that the ball moves in a straight-line. To him, the ball is just as moving in the inertial reference frame and without incorporating fictitious forces, the inertia of the ball can be described by the Newton's Law of Motion. He also find that the ball will miss Thomas. This is because of the ball has inertia to move (without change its direction).
18. Suppose now we take the perspective from Thomas. From his point of view, the ball that Henry throws out is subject to background rotation: at first, it is pointed towards Thomas. With the effect of inertia, the ball keeps moving towards the circumference. But due to rotation, the ball starts to take a re-curving direction and does not point towards Thomas. In the above case, we see the ball reaches the circumference on the right to the Thomas. In this case, we get the same result, but the trajectory of the ball is a curvature, from Thomas's point of view.
19. We can also interpret the re-curving route of the ball like this: Due to inertia of the ball, the ball will continue moving in a straight line in a rotating roundabout. When we stare on the origin, the ball is moving in a straight-line motion. When we are also subject to the background rotation, we will see the ball taking re-curving trajectory. The re-curving trajectory cannot be explained by Laws of Motion. If we want this re-curving trajectory possible to be explained by the Laws of Motion in a simple way, what can we do? What can we do to explain the motion of objects in this frame under Cartesian coordinate, just as in the Inertial Reference Frame?
20. To describe the motion of objects and render laws of physics to be usable in a rotating reference frame just as in the inertial reference frame, a fictitious force is added. To simplify the explanation why it would be in that case, we look at the answer showing the vector on Coriolis force first:
21. To dig out this fundamental concept of force, let's imagine a ball on the plane. Suppose the ball is originally moving forward in straight line. Suddenly, a force to the left is adding on it. Since there is inertia, it will not abruptly turn to the left immediately. Rather, the force will slightly and continuously modify the track. Eventually, if we wait long enough, the original forwarding inertia will become negligible and the ball will turn left.
22. With reference to the case 2b, the ball is originally pointing towards Thomas straight. However, by observation, the ball is deflected to the right, contradicting to the law of inertia (Newton's First Law). Since the law of inertia cannot explain deflection in a rotating reference frame, we add a fictitious force to rescue the law of inertia, so that it can still be used in a rotating reference frame.
23. According to the previous rule regarding modification of track by an additional force, the ball starts deflecting to the right because of more and more "force" has been continuously added to drive the ball turning right. We give this force a name, called Coriolis Force.
24. In the above example, we assumes the speed of rotation is 45 degrees in 1 minute. What happens if the roundabout rotates faster, e.g. 90 degrees in 1 minute?
25. Similarly, we can deduce the track when it rotates 180 degrees in 1 minute.
26. We can compare the result in case 1, 2 and 3, from the perspective of Thomas.
27. If the ball is moving really fast (think it as fast as sound or light), we can intuitively deduce that the ball is almost not subject to the deflection and hence Coriolis Force. Therefore, Coriolis Force is proportional to the speed of rotation and inversely proportional to the speed of the ball.
28. Accordingly, we can also apply this fundamental concept to understand the direction of wind. Just imagine Henry is standing on the North Pole and Thomas sitting on the Equator. We use infinitesimal small size of air particles to substitute the ball. Deflection will still occur and the track of those tiny particles (as known as air masses) will be affected by Coriolis Force as well.
29. If we use another angle to view the Earth, we can find the reason why in Northern Hemisphere, the polar air masses come from the North-east direction but not straightly from the North direction.
30. How about in South Hemisphere? In the South Hemisphere, winds are deflected to the left instead of to the right. Why this will happen?
1402 - 地轉偏向力
"地轉偏向力"好像只是"中文"才有。英文我只聽過Coriolis Force (科里奧利力)。
1. 上回說到地球自轉會影響到風向和極地氣團的流動。以最簡化的方式來說: 氣團從極地吹出會首先往右偏向。假若要了解科氏力的基本成因和由來, 我們不得不利用物理的概念。(這裡我不會把令人拒氣象及物理於門外的一大堆數學公式擺出一副說教的姿態來!)
2. 科氏力必然跟物理上"參考系"的概念有關。以下的例子(不是我發明的) 可以很簡單地描述"參考系"這個概念。
3. 現在假設有一個地鐵月台。Henry 坐在的列車現正以每小時100公里的速度向西移動。Thomas 現在正站在月台上。
4. 從Thomas 的角度而言, 他會發現自己是站著不動 (囧, 不難發現吧!), Henry 則是以每小時100公里的速度向西移動 (即使Henry 坐著列車上不動, 他在Thomas的眼中是正在以高達每小時100公里移動的傢伙!), 跟列車的速度相同。
5. 從Henry 的角度而言, 他會發現自己是坐著不動, 而Thomas 和月台則以每小時100公里的速度向東移動。對於Henry 來說, 那個月台彷彿能夠就像列車一樣高速地逆向移動。
6. 這裹, 我特別地把Thomas 和 Henry 的角度區分出來。在物理上, 這是必要的, 因為Thomas 和 Henry 他們二人有著不相同的參考系。Thomas 是以月台作為其"參考系", 他是以"月台"的角度去看所有東西。而Henry 是以列車作為其"參考系", 所以會把列車作為中心去了解其他事物。(即使你認為月台不能移動, 現在我認為月台是可以移動的, 你不能罵我錯)。 所以你也可以說Thomas 是移動或停留, 視乎你是從那個"參考系" 去看事物。
7. 我們把有以下特性的參考系, 統一地稱為"慣怔參考系"
- 勻速 (速度沒有改變)
- 直線移動
所以, 以上的例子是描述在慣性參考系內物件的運動。慣性參考系必須同時擁有這兩種特性。假如缺少了其中一種特性, 這個參考系便稱為"非慣性參考系"。
8. 牛頓運動第一定律把慣性定義為:
" 當沒有任何外力施加的情況下, 一個物件會避免改變其自身的運動速度, 無論在該物件處於靜止或是運動的狀態下。"
9. 所以, 一輛處於靜止狀態下的車不會無緣無故往前進。另一方面, 一輛在公路上行走的貨車也不會無緣無故突然減速。(對, 就連減速也不會, 更遑論把它停下來)。當年牛頓指出這個匪夷所思的概念曾遭到很大的衝擊: 因為地面上移動的物件會隨時間的流逝而減速, 最後會自然地停下來。在那個年代, 是很難把磨擦力和物件運動分開來看: (1) 物件減速是因為磨擦, 而不是其慣性消失。 (2) 必定要受外力影響才可使物件減速, 慣性令物件不會貿然減速。
10. 在慣性參考系下, 所有運動定律都很容易便能夠在無須任何修正下, 直接地解釋物件的運動。故此, 物理學家引用笛卡兒坐標系統來描述一切物件的運動, 如以下例子:
從Henry 的角度來看
Thomas 現在正逆向移動
有一艘船在緩慢地行駛
另一列地鐵是靜止不動
有個氣球正向Henry 飄來
這些物件假如沒有外力 (外在環境所給予的力), 它們必定會以勻速直線的方式前進, 直到永遠。
11. 地球是一個"旋轉參考系" (是"非慣性參考系"的一種)。這是因為地球圍繞傾斜23.5度的地軸旋轉。
12. 當需要把牛頓運動定律應用到非慣性參考系的時候 (如"旋轉參考系"), 我們必須假設一種假想力, 使運動定律仍然能夠適用於解釋和預測物件移動, 就正如把這些定律應用到慣性參考系的時候一樣。
13. 所以, 我們可以把假想力視為一種額外的力, 以更方便地描述非慣性系內物件的運動方式。這種力也對三個牛頓運動定律提供了補救 , 使這些定律可以適用於非慣性系上。
14. 現在假設Henry 坐在一個巨型的氹氹轉的中心點 (小時候玩過吧), 向邊界推出一個球。現在假設Thomas 正坐在氹氹轉的邊界上, 打算接著Henry 推過來的球。假設現在氹氹轉本身是靜止, 非轉動的。 在無外力的情況下, 由於慣性的關係, 球必定是以勻速直線運動的。不論從Henry 或是 Thomas 的角度, 他們都會一律認同球是以直線的方式移動。這是球在慣性參考系移動下的描述, 不需加入假想力的元素去描述該球的運動。
15. 假設現在有一陌生人讓氹氹轉開始以逆時針的方式轉動 (每分鐘移動45度)。由於Henry 坐著中心點, 無論轉動速度有多大, 他都不會移動的。另一方面, 由於Thomas 坐在邊界上, 他會感到自己的位置改變了 (囧, 也不難發現吧!)。
16. 從Henry 的角度, 他會發覺球仍然以直線的方式運動, 就如在慣性參考系一樣, 所以他並不需要在解釋物體的慣性運動時加入假想力的元素。 除此以外, 他還會發覺Thomas未能把球接到手, 因為他的位置改變了。
17. 從Thomas 的角度而言, 他會發覺那個球會受到氹氹轉的旋轉影響: 起初, 球是向Thomas 的方向移動。由於慣性的關係, 該球會繼續移動直至氹氹轉的邊界。但是, 因為旋轉的關係, Thomas 會發現那個球開始轉向, 並離自己愈來愈遠。於是, 對於Thomas 而言, 那個球的軌跡是一條曲線。
18. 我們也可以把這個現象如此演繹: 由於慣性的關係, 那個球會以直線的方式在旋轉的"氹氹轉"上移動。假如我們凝視著中心點, 我們會發覺該球的確以直線的方式移動, 但是, 如果我們站在Thomas 身後, 跟他一起旋轉, 我們便會發覺那個球是以曲線移動。
19. 牛頓運動定律並不能解釋和描述曲線運動 (因為這是違反慣性的)。那麼, 究竟還有甚麼方法簡單地解釋這種現象, 以及精準地在笛卡兒坐標系統描述這種曲線運動?
20. 為了讓運動力律能夠適用於旋轉參考系上, 我們需要添加一種假想力。(下圖中的箭咀代表力的方向)
21. 我們可以透過以下的例子去深入了解力的特性。假設有一個球體, 球體現在向前移動。在沒有外力的情況下, 該球體在慣性下將會永久地以勻速直線的方式移動。假設有一股引力令球開始朝向右方移動, 那個球絕不會突然立即改變方向, 而是慢慢減速, 逐漸地增加其向右的份量才是。當一段時間過去後, 球才會完全向右方移動。
22. 現在再運用這個概念去重新理解第2個情況中球的軌跡:
23. 我們給予這種力一個名子, 名為科氏奧利力。
24. 再假如上述例子中的氹氹轉不是每分鐘轉45度, 而是轉90度, 那麼球的軌跡又如何?
25. 同樣, 我們也可用直覺推出每分鍾轉180度時的情形。
26. 再把三種情形比較一下, 我們可以得出以下結果。
27. 假如那個球的移動速度很快, 可以想像到在旋轉的平面上, 它幾乎不受這種假想力所影響。所以, 科氏力與物件的運動速度成反比, 與背景旋轉的速度成正比。
28. 同樣地可以把這個概念應用到風向上。與之前的例子一樣, 只要假設Henry 是坐在北極的極點, 而Thomas 坐在赤道上, 並且把球幻想為一些細小的空氣粒子即可導出結論。
29. 如果用第二個角度去觀看, 我們就可以導出為何北極吹來的風都不是從正北方吹來, 而是從東北方向吹來。
30. 那麼, 南半球的情況又是怎樣? 下回再續。
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