HKDSE Physics Core
Chapter 7: Circular Motion | 圓周運動
Billy Sir’s Smart Notes: Master Angular Velocity, Centripetal Force, and Daily Applications.
由中大物理系碩士 Billy Sir 編寫，助你極速掌握角速度、向心力及日常圓周運動應用。
1. Angular Quantities | 角物理量
In circular motion, we use angular quantities to describe how fast an object is rotating. 在圓周運動中，我們使用角物理量來描述物體旋轉的快慢。
| Quantity (物理量) | Definition (定義) | Formula / Unit (公式 / 單位) |
|---|---|---|
| Angular Displacement (\( \theta \)) 角位移 |
The angle turned through by an object in circular motion. (物體在圓周運動中轉過的角度) | Unit: radian (rad) 弧度 |
| Angular Velocity (\( \omega \)) 角速度 |
Rate of change of angular displacement. (角位移的變化率) | $$ \omega = \frac{\Delta \theta}{\Delta t} $$ Unit: rad s\(^{-1}\) |
| Period (\( T \)) & Frequency (\( f \)) 週期與頻率 |
\( T \): Time for one complete revolution. \( f \): Number of revolutions per second. |
$$ T = \frac{1}{f} \quad \text{and} \quad \omega = \frac{2\pi}{T} = 2\pi f $$ |
2. Relationship with Linear Quantities | 與線物理量的關係
Even though an object moves in a circle, it still has a linear speed (tangential speed) along the edge of the circle. 即使物體作圓周運動，它沿著圓的邊緣仍然具有線速度（切線速度）。
📏 Conversion Formulas | 轉換公式
If an object moves in a circle of radius \( r \):
- Arc Length (Distance): \( s = r \theta \)
- Linear Speed: \( v = r \omega \)
3. Centripetal Acceleration & Force | 向心加速度與向心力
An object moving in a circle is constantly changing direction, which means its velocity is changing. Therefore, it must be accelerating towards the center. 作圓周運動的物體不斷改變方向，這意味著其速度在改變。因此，它必定朝向圓心加速。
🎯 Centripetal Acceleration (\( a \)) | 向心加速度
🧲 Centripetal Force (\( F_c \)) | 向心力
According to Newton’s Second Law (\( F = ma \)), the net force required to keep an object in circular motion is the centripetal force.
⚠️ Centripetal vs. Centrifugal Force | 向心力 vs 離心力
Centripetal Force (向心力): The real net force pointing towards the center of the circle (e.g., tension, friction, gravity). (指向圓心的真實淨力，如張力、摩擦力、重力。)
Centrifugal Force (離心力): A fictitious (fake) force felt by an observer inside the rotating frame. It feels like you are being pushed outward, but in reality, it is just your body’s inertia trying to keep you moving in a straight line! (旋轉參考系內的觀察者感受到的「假想力」。感覺被向外推，但實際上只是身體的慣性試圖保持直線運動！)
4. Daily Examples of Circular Motion | 圓周運動的日常例子
In exam questions, you must identify which force provides the centripetal force. 在考試題目中，你必須找出是哪種力提供了向心力。
| Scenario (情境) | Source of Centripetal Force (向心力來源) | Key Equations (關鍵方程) |
|---|---|---|
| 1. Conical Pendulum 圓錐擺 |
Horizontal component of Tension (\( T \sin \theta \)) 張力的水平分量 |
Vertical: \( T \cos \theta = mg \) Horizontal: \( T \sin \theta = \frac{mv^2}{r} \) |
| 2. Skidding of a Car (Flat Road) 汽車轉彎 (平坦路面) |
Friction between tires and road (\( f \)) 輪胎與路面之間的摩擦力 |
\( f = \frac{mv^2}{r} \) Max speed: \( v = \sqrt{\mu g r} \) |
| 3. Banking of Road 傾斜路面轉彎 |
Horizontal component of Normal Force (\( N \sin \theta \)) 法向力的水平分量 |
Vertical: \( N \cos \theta = mg \) Horizontal: \( N \sin \theta = \frac{mv^2}{r} \) \( \tan \theta = \frac{v^2}{rg} \) |
| 4. Turning of an Airplane 飛機轉向 |
Horizontal component of Lift Force (\( L \sin \theta \)) 升力的水平分量 |
Vertical: \( L \cos \theta = mg \) Horizontal: \( L \sin \theta = \frac{mv^2}{r} \) |
Want to get 5** in HKDSE Physics?
Join Billy Sir’s class for full notes, past paper analysis, and exam techniques.
Contact Billy Sir | 聯絡我們