📚 Free-Body Diagrams: How to Draw and Apply Them | 受力分析图：画法与应用
A free-body diagram (FBD) is a simplified sketch showing a single object and all the forces acting on it. It is the single most important tool in mechanics, because it turns a messy physical situation into a clean vector problem. Once the diagram is correct, the mathematics becomes routine.
受力分析图（Free-Body Diagram, FBD）是一种简化示意图，只画出一个物体以及作用在它上面的所有力。它是力学中最重要的工具，因为它能把复杂的物理情境转化为清晰的矢量问题。只要图画对了，数学计算就变得有条不紊。
1. What Is a Free-Body Diagram? | 什么是受力分析图？
A free-body diagram isolates one object from its surroundings. We draw the object as a dot or a simple box, then draw arrows representing every external force acting on it. Internal forces are not included, and forces the object exerts on other objects are not included.
受力分析图把某个物体从周围环境中“隔离”出来。我们将物体画成一个点或一个简单的方框，然后用箭头画出所有作用在它上面的外力。内力不画进去，物体对其他物体施加的力也不画进去。
The standard symbols are simple:
标准的符号约定很简单：
- W or mg — weight (always downward, toward the centre of the Earth)
- N or R — normal reaction (always perpendicular to the contact surface)
- T — tension (always pulling away from the object along a string or rod)
- F — applied force (any external push or pull)
- f or Fkr — friction (parallel to the contact surface, opposing motion or tendency to move)
- W 或 mg — 重力（总是竖直向下，指向地心）
- N 或 R — 法向反力（总是垂直于接触面）
- T — 拉力（总是沿绳或杆离开物体）
- F — 外加力（任何外部推或拉）
- f 或 Fkr — 摩擦力（平行于接触面，阻碍运动或运动趋势）
Each arrow must show the correct direction, and ideally the relative magnitude. The point of application matters when moments are involved, but for pure translation problems, a dot is sufficient.
每个箭头都必须方向正确，最好还能大致反映力的相对大小。涉及力矩时，作用点的位置很重要；但如果是纯粹的平动问题，画成一个点就足够了。
2. Why Are Free-Body Diagrams So Important? | 为什么受力分析图如此重要？
can only be applied correctly if you know every force acting along each axis. Drawing an FBD forces you to account for each force explicitly, preventing the two most common errors: missing forces and inventing forces that do not exist.
力学问题本质上是矢量方程。只有当你清楚每个坐标轴方向上的所有力时，才能正确运用 F = ma。画受力分析图迫使你把每个力都明确列出来，从而避免两大常见错误：漏掉力，或凭空多造出力。
A well-drawn FBD also reveals the physics. For example, an object resting on a slope has its weight resolved into components, which immediately explains why a steeper slope means a larger downhill component. The diagram is not a formality — it is the analysis itself.
一张好的受力分析图还能揭示物理本质。例如，斜面上静止的物体，其重力需分解为两个分量，这立即解释了为什么斜面越陡，沿斜面向下的分量越大。受力分析图不是走形式——它本身就是分析过程。
3. The Key Forces You Must Know | 必须掌握的基本力类型
Before drawing anything, you must be able to recognise the forces that appear at contact points and at a distance.
在动手画图之前，你必须能识别出现在接触点和超距作用中的各种力。
Weight (W = mg) always points vertically downward, from the object's centre of mass. It does not change direction when the object is on a slope. A common mistake is to draw weight perpendicular to the slope, but that is wrong: weight is always vertical.
重力（W = mg）始终竖直向下，作用于物体的重心。即使物体在斜面上，重力的方向也不会改变。常见的错误是把重力画成垂直于斜面，这是不对的：重力永远竖直向下。
Normal reaction (N) is the contact force from a surface, always perpendicular to that surface. On a horizontal floor, N is vertical; on a slope, N is perpendicular to the slope. It is not always equal to mg — it adjusts to satisfy equilibrium or acceleration in the perpendicular direction.
法向反力（N）是接触面施加的力，总是垂直于接触面。在水平地面上，N 竖直向上；在斜面上，N 垂直于斜面。N 不一定等于 mg——它会根据垂直方向的平衡或加速度情况自行调整。
Tension (T) acts along the string, rope or rod, always pulling away from the object. A string cannot push. If a string passes over a pulley, the tension is usually the same on both sides (assuming a light, frictionless pulley).
拉力（T）沿绳、索或杆的方向作用，总是拉着物体离开。绳子不能推。如果绳子绕过定滑轮，在轻质无摩擦滑轮的前提下，两侧拉力通常相等。
Friction (f) to move. Static friction satisfies f ≤ μ_s N, while kinetic friction satisfies f = μ_k N.
摩擦力（f）平行于接触面，阻碍相对运动或运动趋势。静摩擦力满足 f ≤ μ_s N，而动摩擦力满足 f = μ_k N。
4. The Six-Step Method for Drawing an FBD | 画受力分析图的六步法
The following procedure works for every mechanics problem. Follow it strictly, and you will never miss a force.
下面的步骤适用于所有力学问题。严格按步骤走，你永远不会漏掉力。
Step 1 — Choose the object. Circle it mentally. Everything outside that circle is the environment; the environment acts on the object, never the other way around.
第一步——选定研究对象。在心里把它圈出来。圆圈之外的一切都是环境；是环境对物体施加力，而不是反过来。
Step 2 — Draw the object. Draw a dot or a box. Do not draw the surface, the strings, or other objects — they are not part of the diagram.
第二步——画出物体。画一个点或一个方框。不要画接触面、绳子或其他物体——它们不属于受力分析图。
Step 3 — Draw weight. Start with mg, vertically downward. This is the only force that never changes direction.
第三步——画重力。先画 mg，竖直向下。这是唯一方向永远不变的力。
Step 4 — Identify contacts. For each point where the object touches another body, ask: can this contact push or pull? Draw the normal reaction perpendicular to the surface, and the tension along any string.
第四步——找接触点。对物体与外界接触的每一个点，问自己：这个接触能推还是能拉？画出与接触面垂直的法向反力，画出沿绳方向的拉力。
Step 5 — Add friction. If there is motion or an applied force along a surface, friction acts in the opposite direction. If unsure, draw it and later check the sign of the answer.
第五步——加上摩擦力。如果物体沿接触面运动或受到沿接触面的外力，摩擦力方向与之相反。如果不确定方向，先画出来，最后通过答案的正负号来检验。
Step 6 — Label and resolve. Label each arrow with its symbol. Then choose axes (usually horizontal/vertical, or along/perp to the slope) and resolve any force that is not parallel to an axis.
第六步——标注并分解。给每个箭头标上符号。然后选择坐标轴（通常取水平/竖直，或沿斜面/垂直斜面），将不与坐标轴平行的力进行分解。
5. Choosing Axes and Resolving Forces | 选择坐标轴与力的分解
Resolving a force means splitting it into two perpendicular components. For a force F at an angle θ to the x-axis:
力的分解是指把一个力拆成两个相互垂直的分量。设力 F 与 x 轴夹角为 θ：
Fₓ = F cos θ, Fy = F sin θ
The choice of axes should match the motion. If the object slides along a plane, take one axis along the plane and the other perpendicular to it. This way, the acceleration has only one component, and the perpendicular equation usually yields the normal reaction.
坐标轴的选择应与运动方向相匹配。如果物体沿某平面滑动，就让一个轴沿平面方向，另一个轴垂直于平面。这样一来，加速度只有一个分量，且垂直方向的方程通常能直接解出法向反力。
On a slope of angle θ, the weight components are:
在倾角为 θ 的斜面上，重力的分量为：
W_∥ = mg sin θ (down the slope), W_⊥ = mg cos θ (into the slope)
al and the normal to the slope. Always check your diagram with a simple case: if θ = 0, W_∥ must be zero; if θ = 90°, W_∥ must be mg.
注意角度的位置：斜面与水平面的夹角，正好也等于竖直方向与斜面法向的夹角。务必用特例检验你的图：若 θ = 0°，W_∥ 应为零；若 θ = 90°，W_∥ 应等于 mg。
6. Example: Particle on a Horizontal Surface | 实例：水平面上的物体
A box of mass 10 kg is being pulled by a force of 40 N at 30° above the horizontal. The coefficient of friction is 0.2. Find the acceleration.
一个质量 10 kg 的箱子受到与水平方向成 30° 角、大小为 40 N 的拉力。摩擦系数为 0.2。求加速度。
The FBD has four forces: weight mg downward, normal reaction N upward, applied force F at 30°, friction f opposing motion. Since F is not along an axis, resolve it:
受力分析图上有四个力：重力 mg 向下、法向反力 N 向上、外加力 F（与水平成 30°）、摩擦力 f（阻碍运动）。由于 F 不沿坐标轴方向，需要分解：
Fₓ = 40 cos 30° = 34.6 N, Fy = 40 sin 30° = 20 N
Vertical equilibrium:
竖直方向平衡：
N + Fy = mg → N = 10 × 9.8 − 20 = 78 N
Friction:
摩擦力：
f = μ N = 0.2 × 78 = 15.6 N
Horizontal equation:
水平方向方程：
Fₓ − f = ma → 34.6 − 15.6 = 10a → a = 1.9 m s⁻²
The upward component of the applied force reduces the normal reaction, which in turn reduces friction. This is the key insight the FBD reveals at a glance.
拉力向上的分量减小了法向反力，从而也减小了摩擦力。这正是受力分析图一眼就能看出的关键结论。
7. Example: Particle on a Smooth Slope | 实例：光滑斜面上的物体
A particle of mass 5 kg slides down a smooth plane inclined at 25° to the horizontal. Find its acceleration and the normal reaction.
一个质量 5 kg 的质点沿倾角 25° 的光滑斜面下滑。求加速度和法向反力。
The FBD has only two forces: weight mg vertically down, and normal reaction N perpendicular to the plane. Resolve weight along and perpendicular to the slope:
受力分析图只有两个力：重力 mg 竖直向下，法向反力 N 垂直于斜面。将重力沿斜面方向和垂直斜面方向分解：
W_∥ = mg sin 25° = 5 × 9.8 × 0.4226 = 20.7 N
W_⊥ = mg cos 25° = 5 × 9.8 × 0.9063 = 44.4 N
Perpendicular to the slope, there is no acceleration:
垂直斜面方向无加速度：
N = W_⊥ = 44.4 N
Along the slope:
沿斜面方向：
W_∥ = ma → a = 20.7 ÷ 5 = 4.14 m s⁻²
mg cos θ. Drawing the correct FBD prevents the classic error of writing N = mg on a slope.
注意法向反力并不等于 mg，而是 mg cos θ。画出正确的受力分析图可以避免“在斜面上写 N = mg”这一经典错误。
8. Systems and Newton's Third Law | 物体系统与牛顿第三定律
When two objects are connected by a string or touching each other, each object must have its own FBD. The action-reaction pair is drawn on separate diagrams, not on the same one.
当两个物体通过绳子相连或相互接触时，每个物体都必须有自己独立的受力分析图。作用力与反作用力要分别画在不同的图上，绝不能画在同一张图上。
For example, if a 2 kg box pushes against a 3 kg box with a force P, the second box pushes back on the first with force P in the opposite direction. On the FBD of the first box, the force is drawn pointing left; on the FBD of the second box, the same force is drawn pointing right. They have the same magnitude but opposite directions — that is Newton's third law.
例如，若一个 2 kg 的箱子以力 P 推一个 3 kg 的箱子，那么第二个箱子也会以大小相同、方向相反的力 P 推第一个箱子。在第一个箱子的受力分析图上，该力指向左；在第二个箱子的图上，该力指向右。两者大小相同、方向相反——这就是牛顿第三定律。
When solving connected-body problems, one common strategy is to treat the whole system as a single object to find the acceleration, then use an individual FBD to find internal forces like tension. For a system of total mass M with external force F:
在求解连接体问题时，一种常用策略是先把整个系统看作一个整体来求加速度，再通过单个物体的受力分析图来求内力（如绳中张力）。设系统总质量为 M，受外力 F：
a = F / M (for the whole system)
e the tension. The FBD makes it clear which forces are external to the system and which are internal.
然后对其中一个物体列 F = ma 方程，即可解出张力。受力分析图能清晰区分哪些力对系统是外力、哪些是内力。
9. Tension Problems: Hanging Masses | 悬吊问题：悬挂物体
A mass hangs from two strings, one horizontal and one at 40° to the vertical. The FBD shows three forces: weight mg downward, tension T₁ in the horizontal string, and tension T₂ in the angled string. Since the particle is at rest, the vector sum is zero.
一个质量用两根绳悬挂，一根水平，一根与竖直方向成 40° 角。受力分析图上有三个力：重力 mg 向下、水平绳中的张力 T₁、斜绳中的张力 T₂。因质点静止，合力为零。
Resolving horizontally and vertically:
分别沿水平和竖直方向分解：
T₂ sin 40° = T₁ (horizontal)
T₂ cos 40° = mg (vertical)
os 40°, which is greater than mg. This means the angled string carries more than the weight. Intuition might suggest otherwise, but the FBD plus trigonometry settles it immediately.
由竖直方程得 T₂ = mg / cos 40°，大于 mg。这说明斜绳所受的力比重力还大。直觉可能恰好相反，但受力分析图结合三角计算能立刻给出答案。
10. Common Mistakes and the Final Checklist | 常见错误与最终检查清单
Every year, students lose marks on mechanics not because of weak algebra, but because of careless FBDs. Here are the most frequent errors.
每年都有学生在力学题上失分，原因往往不是代数太弱，而是受力分析图画得太马虎。以下是最常见的错误。
Mistake 1: Forgetting weight. Weight acts on every object near the Earth's surface. It is never optional.
错误一：漏画重力。地球表面附近的每个物体都受到重力，它绝不是可画可不画的。
Mistake 2: Drawing forces on the wrong object. The FBD must only show forces acting on the chosen object, not forces it exerts on others. This is the most systematic error in mechanics.
错误二：把力画错对象。受力分析图只能画出作用于所选物体上的力，不能画出它施加给其他物体的力。这是力学中最系统性的错误。
Mistake 3: Assuming N = mg. On a slope or when a vertical component of another force exists, N is different from mg. Resolve first; do not assume.
错误三：默认 N = mg。在斜面上，或者当其他力有竖直分量时，N 并不等于 mg。先分解，再计算，不要想当然。
Mistake 4: Friction always opposing motion. Friction opposes relative motion or the tendency to move. If the applied force is not enough to move the object, friction acts in the opposite direction to the applied force, not necessarily opposite to velocity (because velocity is zero).
错误四：认为摩擦力总是阻碍运动。摩擦力阻碍的是相对运动或相对运动趋势。当外力不足以推动物体时，静摩擦力的方向与外力方向相反，而不是与速度方向相反（因为此时速度为零）。
Mistake 5: Mixing up sin and cos. along the slope is mg sin θ; in a hanging-mass problem, the vertical component of tension is T cos θ. Write the angle explicitly on your diagram to avoid confusion.
错误五：混淆 sin 和 cos。始终用 θ = 0° 和 θ = 90° 两个特例来检验。在斜面上，沿斜面方向的分量是 mg sin θ；在悬挂问题中，张力的竖直分量是 T cos θ。在图上明确标出角度，避免混乱。
Final checklist before writing equations:
列方程前的最终检查清单：
- Is the FBD drawn for exactly one object?
- Is weight included and vertical?
- Are all contact forces present, each perpendicular to its surface?
- Are all tensions pulling along their strings?
- Is friction parallel to the surface and opposite to motion or motion tendency?
- Have all non-axial forces been resolved into components?
- Have you checked limiting cases θ = 0° and θ = 90°?
- 受力分析图是否只针对一个物体？
- 重力是否已画出且竖直向下？
- 所有接触力是否都已列出，且垂直于各自接触面？
- 所有拉力是否都沿绳方向向外？
- 摩擦力是否平行于接触面且与运动或运动趋势方向相反？
- 所有不沿坐标轴的力是否都已分解？
- 是否用 θ = 0° 和 θ = 90° 检验过极限情形？
11. Putting It All Together | 综合应用
The power of a free-body diagram lies in its ability to convert a physical scenario into a clean mathematical one. Once the diagram is correct, the equations follow mechanically: resolve, write Newton's second law along each axis, and solve. The hardest part of any mechanics problem is not the algebra — it is drawing the correct diagram in the first place.
受力分析图的威力在于，它能把物理情境转化为清晰的数学问题。一旦图画对了，列方程就是机械操作：分解、沿各轴写牛顿第二定律、求解。任何力学题最难的部分都不是代数——而是第一步画对图。
Practise every day with a new scenario: a block on a ramp, a ladder against a wall, a satellite in orbit, a car rounding a curve. In each case, always start from the same six-step method. Over time, the process becomes automatic, and the subject becomes much easier.
每天都用新的场景来练习：斜面上的滑块、靠墙的梯子、轨道上的卫星、转弯的汽车。无论哪种情况，都从同样的六步法开始。久而久之，这个过程会变成一种本能，而力学也会变得轻松许多。
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