費瑪也瘋狂-平面上存在障礙時連接三定點的最佳網絡問題
在一個有障礙的平面上，給三個定點，我們探討連接此三點的最佳網絡。我們討論了諸如直線、射線、線段、圓、網格狀、三角形……等類的障礙，當網絡每穿越障礙一次，就必須付出代價，例如「拖延5 分鐘」。所以，設網絡穿越障礙的次數為y ，則網絡除了原本的總長度之外，還額外加入y 倍某固定數值的損耗。我們以費瑪點的各種性質及三角形不等式等方法為工具，就不同的穿越障礙次數綜合比較，而找出最佳網絡。在某些情況下，最佳網絡不是以費瑪點來連接三點，而是在障礙(如：直線)上找出符合某種與餘弦值相關特殊性質的點，以該點來連接三點，而此網絡可用GSP 軟體相當精確地作出。另外，我們也探討在考慮障礙造成損耗的情況下，兩點間的「實際距離」為何。 最後，我們考慮「混合障礙」問題。在此類問題中，除了前面所討論的障礙，還另加了如同「河流」的兩平行直線間區域之障礙，在這種障礙區域中，網絡的長度要乘以數倍來計算。我們發現，此類問題的最佳網絡也可用特定的正弦條件配合GSP 而相當精確地作出來。;Considering various kinds of obstacles in a plane, such as a line, a segment, a ray, a circle, a triangle or chessboard grids, which function like a red light, we research into the problem of finding the optimal network connecting three given points A, B, C in the plane amidst obstacles described above. Each time when the network crosses an obstacle, it will cause losses, such as five minute’s delay or a loss of one hundred dollars. Taking advantage of Fermat points, some basic inequalities concerning triangles and some special qualities about sine or cosine functions, we obtain the optimal networks in different situations. Besides, we consider what the “real distance” between two points is when there are obstacles in a plane. We also put another obstacle, including a line and a weighted region between two parallel lines, into consideration. In the region, like a river or a muddy ground in real life, the length of the network should be multiplied by a fixed time. Furthermore, we can use GSP to make the networks very accurately.
鄒之風聲-風笛
「風笛」是台灣原住民鄒族的信號用具及祈雨法器，由一條繩子綁一支竹片構成。轉動風笛時，竹片會繞繩子自轉並拍打空氣而發出聲音，並有上下飛舞的現象。風笛產生聲音的原因，為竹片拍打空氣而造成的渦流共振現象；又由於繩子扭力大小及方向改變，使風笛的音調忽高忽低、響度忽大忽小、且竹片會在兩個平面上公轉，而有週期性變化。施力使風笛公轉轉速加快時，竹片自轉速率也變快，使其音調愈高、響度愈大；而繩愈短、愈粗時，竹片的公轉週期將愈短。The wind whistler was once used by Tsou aborigines as a tool for message transfer. It is composed of a string and a bamboo flapper. When swung around, the flapper spins, beats the air, and makes sounds. Moreover, the flapper flies up and down during the revolution. The spinning flapper beats the air, causes the vortex resonance phenomenon, and thus produces sound. As the twist torque and direction change, there is periodical variation in the sound volume, sound pitch, and the movement of the flapper, which orbits up and down at two planes. If given force to speed up its revolution, the flapper，s spinning frequency also increases, which makes the sound pitch higher and the sound volume greater. Besides, when the string is shorter or thicker, the flapper，s revolution period will be shorter.
金屬的盔甲
Our aim to attend this science fair is to design an instrument that can plat and measure the mass at the same time. In hope of designing a simple, accurate and convenient apparatus, we created an electronic circuit to display our original idea. In the process of constant improvements, we finally accomplished a “Super Mass Plating Gauge”, which can be easily and widely utilized in school teaching. The production of microbalance and the arrangement of electric circuit are the most significant parts in our research. The major components of the “Super Mass Plating Gauge” include a straw, metal clips and our creativity—the well-arranged electric circuit. The idea of microbalance originated from the Internet, but we advanced it by numerous experiments. First, we attached a steel cord to one side of the cathode in the electricity supplier. Next, we fixed the other side to the negative plate. And then, on the end of the negative plate, we tied a metal clip with the metal that will be plated. Eventually a new “plating gauge” was invented. By doing so, we could use this instrument to make our experiments. Our experimental goal is to research how different kinds of metal, time, electrode and voltage can affect the reduced mass on the cathode. We made use of such metal as copper, zinc and silver to carry out the experiments. In the end, by analyzing the results, we concluded a plating formula that can be applied to metal plating.
我們做此科展的目的，是要設計一個可以邊電鍍、邊測量質量的儀器，我們希望這個儀器是簡便、精確、且線路簡單，並且能推廣到教學的器材。經過我們不斷改良，終於完成了「便利質量電鍍器」 。 其中製作微量天秤和線路的配置方法，是本研究的重要部分。微量天秤的主要結構是吸管、鱷魚夾、及線路。微量天秤的構想，是參考以前的科展作品並加以改良，可精準測量到0.00010g，而裡面的線路，則是我們的創意（如圖一） 。只要把電源供應器的正極，接上左右任一鋼條，負極接到容器另一端，並加上一個鱷魚夾，夾上被鍍物，便是一個可邊電鍍，邊測量質量的儀器了！如此一來，我們就能以此儀器來作我們以下的實驗。 我們實驗目的在探討電鍍時不同金屬、不同時間、電極大小及電壓，對正極金屬片所減少質量的影響。 最後，我們推導出一個有關電鍍時正極金屬片質量變化量的實驗公式。為此，我們也要做許多次、許多種的實驗，來驗證我們的公式是否正確，並以我們所學的理論來推論。
毛細管內液體流速之探討及黏度測量方法之創新
有一次上實驗課時看到鄰桌的同學正以實驗室的毛細管吸吮一瓶\r 葡萄汁，每當他吞嚥時，毛細管內殘餘的葡萄汁竟然以近似等速的狀\r 態降回果汁瓶裡，這似乎違背了重力加速度的常理，是毛細管的影響\r 嗎？還是葡萄汁特有的現象？這與毛細管插入葡萄汁的深度有關嗎？\r 這一連串的問題皆因升學壓力而無暇深究。\r 升上高中的那年暑假，我參加了學校舉辦的資優科學營，在物理\r 實驗課中恰好遇到了以毛細管測量液體黏度的實驗，使我有機會深入\r 思索毛細管內液體流速的問題。這個實驗很不好做，要調整兩個定高\r 容器，使水平毛細管兩端產生固定的壓差，讓流經管內的液體為等速\r 的流動。大家手忙腳亂地操作了近三小時，測出來的黏度不僅和老師\r 公布的答案差距甚大，而且各組同學所測得之結果也大相逕庭，不過\r 我們都體驗到了液體黏度對流速的影響。\r 實驗的誤差為什麼那麼大？扣除人為操作誤差後儀器本身還有多\r 大改進空間？有否更精簡的方法測量黏度？毛細管中液體的流速真\r 是固定的嗎？把毛細管斜置或是垂直架置，流速還會固定嗎？重力不\r 會影響流速嗎？若流速真為固定的，如何控制或改變流速呢？是否可\r 以利用流速來測出液體黏度？\r 由於我對這個實驗的專注，在學校選修的專題課程中即以此為題\r 目做長期的研究，在老師的指導下，我和同組同學不停地查閱有關書\r 籍及整理資料，漸漸研究清楚原理後定出實驗計畫，並開始購買材料\r 與工具，長期地動手實驗及改進，兩年來我們利用毛細管中液體的等\r 速及不等速流動的特性，發展出了兩種既精簡又準確的液體黏度測量\r 法，以下就是整個研究及創作的過程。\r \r \r The viscosity is one of the most important parameters of the fluids. In conventional viscosity experiments, capillary-based viscometers are widely adopted because of their user-ready devices and moderate prices. However, to accurately measure the viscosity, the fluid velocity must be kept constant; otherwise, the fluctuation may cause the serious deviation. Besides, traditional capillary viscometers often require a long time to make the measurement. This research aims at exploring the capillary fluid mechanics and developing accurate and rapid methods to measure the viscosity. In this study, we developed two creative viscosity-measuring methods, the gravity sloping helical structure and the gradational liquid-level difference, and constructed two cost-effective capillary viscometer prototypes accordingly. The gravity sloping helical structure employs a long helical capillary as the flow channel and utilizes the gravity force to keep the fluid velocity extremely constant. The highlight of this method is that we can see the fluid flow through the capillary directly. On the other hand, the gradational liquid-level difference method uses the pressure sensors to precisely monitor the liquid-level difference, which drives the flow in the capillary, and allows automatic measurement of viscosity. Compared with computer simulation, the experimental results agreed well with the theoretical values. The gravity sloping helical and gradational liquid-level difference methods achieve 0.5% and 0.8% accuracy respectively. In addition, the automation of viscosity sensing also greatly facilitates the viscosity measurement. Furthermore, these two methods are both valuable for incorporating into educational purpose and industrial application.
滑鼠狂想曲
光學滑鼠會以很高的速度不斷地對著接觸面拍照，藉由比對每幅影像間的變化來偵測滑鼠移動的速度與方向，本研究利用此特點而設計一個簡易的光學量測系統，其中包括透鏡、光源與接觸面材質的選擇，以及利用Raw Input 模式讀取個別滑鼠移動訊息而發展出來的量測程式，使得此系統可以在無接觸與無摩擦的情況下來測量外界物體的移動速度與距離，經由實驗證明，在光學感測器還可以感應與追蹤的範圍內，量測的數據還蠻精準的。接觸面到光學感測器透鏡的距離越遠，能夠測得移動物體的極速也越高，但是會造成感測器的解析度下降，如此限制了接觸面的材質種類，無法量測表面較為光滑的物體，但是在設計得宜的情況下，仍有蠻多方面的用途，日後若能採用較高效能的光學感測器並加上測距儀的輔助，相信此系統的應用層面會更為廣泛。Optical mouse can take continuous snapshots very quickly of the contact surface and compare the images sequentially to detect the direction and amount of movement. This study uses this feature to design a simple optical measurement system, including lens, illumination and contact surface choice, as well as the measurement program using raw input model to accept the movement information from the mouse. This system can measure the distance and speed of the motion object under the non-friction condition. From the experiment test result, this optical measurement system is workable and satisfactory. Contact surface to optical sensor distance farther, can measure the higher speed of the motion object, but will cause the lower resolution of the optical sensor. This will limit the variety of the contact surface; superficial smoother object is unable to measure. In the future if we can use the high performance optical sensor and assist with rangefinder, believed this system can have more widespread applications.