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HKDSE Physics – Moment of Force

HKDSE Physics Core
Chapter 3: Moment of Force | 力矩

Billy Sir’s Smart Notes: Master Torque, Equilibrium, Centre of Gravity, and Toppling.
由中大物理系碩士 Billy Sir 編寫，助你極速掌握力矩、平衡狀態、重心及傾倒條件。

1. Torque (Moment of a Force) | 力矩

The turning effect of a force is called its moment or torque. 力產生的轉動效應稱為力矩。

🔄 Moment Formula | 力矩公式

$$ \tau = F \times d $$
  • \(\tau\): Moment of force (Unit: N m) (力矩，單位：牛頓米)
  • \(F\): Applied force (Unit: N) (施加的力)
  • \(d\): Perpendicular distance from the pivot to the line of action of the force (由支點到力作用線的垂直距離)

Billy Sir’s Tip: Always check if the distance given is perpendicular to the force. If the force is applied at an angle \(\theta\), use \(d \sin\theta\).

2. Resultant Torque & Principle of Moments | 淨力矩與力矩原理

When multiple forces act on a body, the resultant torque is the sum of all individual torques, taking direction (clockwise or anticlockwise) into account.

⚖️ Principle of Moments | 力矩原理

For an object to be in rotational equilibrium, the sum of clockwise moments about any point must equal the sum of anticlockwise moments about that same point.

$$ \sum \tau_{\text{clockwise}} = \sum \tau_{\text{anticlockwise}} $$
當物體處於轉動平衡時，作用於物體上任何一點的順時針力矩總和，必定等於逆時針力矩總和。

3. Static Equilibrium | 靜態平衡

Condition (條件) Mathematical Expression (數學表達式) Physical Meaning (物理意義)
Translational Equilibrium
平移平衡		 $$ \sum F = 0 $$ No net force. The object will not accelerate linearly. (Upward forces = Downward forces, Left = Right).
Rotational Equilibrium
轉動平衡		 $$ \sum \tau = 0 $$ No net moment. The object will not undergo angular acceleration.

Note: For an object to be in complete static equilibrium, both conditions must be satisfied simultaneously.

4. Centre of Gravity (CG) | 重心

🎯 Definition | 定義

The Centre of Gravity (CG) is the point through which the entire weight of an object appears to act. 重心是物體全部重量的等效作用點。

  • For a uniform symmetrical object, the CG is at its geometric centre.
  • The CG can lie outside the material of the body (e.g., a ring or a hollow pipe).
  • If an object is suspended freely from a point, its CG will always lie vertically below the point of suspension.

5. Stability and Toppling | 穩定性與傾倒

Whether an object will topple depends on the position of its Centre of Gravity relative to its base of support.

⚠️ Condition for Toppling | 傾倒條件

An object will topple if the vertical line passing through its Centre of Gravity falls outside its base of support. This creates a net overturning moment.

如果穿過重心的垂直線落在支撐面（底座）之外，物體便會傾倒。因為重力會產生一個使其翻倒的淨力矩。
Scenario: Object on an Inclined Plane (斜面上的物體) How to Increase Stability (如何增加穩定性)
As the angle of the incline increases, the vertical line from the CG moves closer to the edge of the base. Once it passes the pivot edge, the object topples.
  • Lower the Centre of Gravity: Place heavier components at the bottom. (降低重心)
  • Widen the Base Area: Increases the angle required for the CG line to fall outside. (加闊底座)

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