力矩、平衡与转动Torque, equilibrium & rotation
力矩与转动在 Section 1 中反复出现。本页选五题，每题 3–6 分，共 22 分：常见失分点是选错力臂、漏掉支持力，或把平衡方程套到有向心加速度的物体上。先画清受力，再决定取矩点和使用的定律。Torque and rotation recur in Section 1. These five questions carry 3–6 marks each, 22 in total. Typical errors are incorrect moment arms, missing reactions, or treating centripetal motion as static equilibrium. Draw the forces first, then choose the moment origin and governing law.
1力矩看垂直距离Use perpendicular moment arms
选取取矩点 P，力的作用线到 P 的垂直距离才是力臂。静止刚体既要合力为零，也要合力矩为零；改取矩点不会改变平衡结果。The arm is the perpendicular distance from P to the force’s line of action. A static rigid body needs both zero net force and zero net torque; changing the moment origin cannot change equilibrium.
优先在未知力的作用点取矩，可以消去这个力。Taking moments where an unknown force acts eliminates its moment.
2重心、浮心与稳定性Mass centre, buoyancy centre and stability
均匀直杆的重心在中点，浮心在浸没体积的几何中心。悬挂物体的重心处于支点正下方时，微小偏转会产生恢复力矩。A uniform rod has its mass centre at the midpoint and buoyancy centre at the centre of displaced volume. A suspended body with its mass centre below the pivot experiences a restoring torque after a small displacement.
重心在支点上方也可能零力矩，但通常是不稳定平衡。A centre of mass above a pivot can give zero torque but unstable equilibrium.
3转动动能与径向受力Rotational energy and radial forces
质点绕固定轴运动时，转动惯量由质量和到轴的距离决定。无摩擦时用能量守恒联系不同位置的速率，再按指向圆心的方向列牛顿第二定律。For a particle about a fixed axle, inertia depends on mass and distance from the axle. Use energy conservation to connect speeds when friction is negligible, then apply Newton’s second law toward the centre.
向心力是合力的径向分量，不是一种额外的力。Centripetal force is the radial net force, not an extra force.
| 子题型Sub-type | 识别特征Recognise | 核心步骤Core step | 真题Question |
|---|---|---|---|
| L 形杠杆L-shaped lever | 两臂垂直Perpendicular arms | 平衡重力矩Balance weight moments | 2024 Q1(e) |
| 墙梯Leaning plank | 光滑墙与地面Smooth wall and floor | 底端取矩Moments about the foot | 2023 Q1(d) |
| 浮杆Floating rod | 部分浸没且悬挂Immersed and suspended | 浮心 + 力矩Buoyancy centre + torque | 2023 Q1(r) |
| 转弯倾斜Lean in a turn | 速度改变、半径不变New speed, same radius | tanθ ∝ v²tanθ ∝ v² | 2022 Q1(c) |
| 竖直圆周Vertical circle | 顶部张力为零Zero top tension | 能量 + 径向方程Energy + radial equation | 2017 Q1(k) |