🪢 张力计算器（竖直加速、斜拉两绳、斜面、滑轮两物体）Tension Calculator (Vertical, Two Angled Ropes, Incline, Pulley With Two Masses)

使用说明与常见问题How-to & FAQ

「张力计算器（竖直加速、斜拉两绳、斜面、滑轮两物体）」是什么？What is the Tension Calculator (Vertical, Two Angled Ropes, Incline, Pulley With Two Masses) tool?

四种常见绳张力情形：竖直吊挂并加速 T = m(g + a)；两根斜绳吊挂一个物体（给出两绳与水平线的夹角）；斜面上被平行于斜面的绳拉住的物体（可含摩擦）；以及跨过定滑轮的两物体（阿特伍德机）的加速度和张力。本地处理。Four common rope tension set-ups: a hanging mass accelerating vertically, T = m(g + a); a mass hung from two angled ropes (angles from the horizontal); a mass on an incline held by a rope parallel to the slope (with optional friction); and two masses over a fixed pulley (Atwood machine) with the acceleration and tension. Processed locally.

如何使用「张力计算器（竖直加速、斜拉两绳、斜面、滑轮两物体）」？How do I use the Tension Calculator (Vertical, Two Angled Ropes, Incline, Pulley With Two Masses) tool?

在「张力计算器（竖直加速、斜拉两绳、斜面、滑轮两物体）」中，填写「情形」、「质量 kg」、「加速度（向上为正）」、「g」、「绳 1 与水平夹角 °」等参数，结果会随输入实时计算并显示在下方。In the Tension Calculator (Vertical, Two Angled Ropes, Incline, Pulley With Two Masses): Fill in "Set-up", "Mass kg", "Accel (up +)", "g", "Rope 1 angle from horizontal °"; results are recalculated instantly as you type.

竖直吊挂的张力怎么算？How is tension found for a hanging mass?

静止或匀速时 T = mg；向上加速 a 时 T = m(g + a)。10 kg 物体静止时张力 98.1 N，以 2 m/s² 向上加速时 118.1 N。At rest or constant speed T = mg; accelerating upward at a, T = m(g + a). A 10 kg mass gives 98.1 N at rest and 118.1 N accelerating up at 2 m/s².

两根斜绳怎么算？How about two angled ropes?

水平分量相等、竖直分量之和等于重力：T₁ = W·cos θ₂ / sin(θ₁ + θ₂)，T₂ = W·cos θ₁ / sin(θ₁ + θ₂)，角度从水平线量。角度越小张力越大。The horizontal parts cancel and the vertical parts add to the weight: T₁ = W·cos θ₂ / sin(θ₁ + θ₂) and T₂ = W·cos θ₁ / sin(θ₁ + θ₂), with angles from the horizontal. Shallower ropes carry more tension.

滑轮两物体（阿特伍德机）Two masses over a pulley (Atwood machine)

加速度 a = (m₁ − m₂)g / (m₁ + m₂)，张力 T = 2m₁m₂g / (m₁ + m₂)，假设绳和滑轮无质量、无摩擦。Acceleration a = (m₁ − m₂)g / (m₁ + m₂) and tension T = 2m₁m₂g / (m₁ + m₂), assuming a massless, frictionless rope and pulley.

能举个计算实例吗？Can you show a worked example?

例如在「张力计算器（竖直加速、斜拉两绳、斜面、滑轮两物体）」中填 质量 kg = 10，加速度（向上为正） = 0，g = 9.81，结果区显示：张力 N：98.1For example, entering Mass kg = 10, Accel (up +) = 0, g = 9.81 in the Tension Calculator (Vertical, Two Angled Ropes, Incline, Pulley With Two Masses) gives: Tension N: 98.1

哪些输入会被检查？Which inputs are validated?

「张力计算器（竖直加速、斜拉两绳、斜面、滑轮两物体）」会检查这些输入——「质量 kg」：必须填写，须大于 0；「加速度（向上为正）」：可留空（留空按 0）；「g」：所选方式用到时必须填写；「绳 1 与水平夹角 °」：所选方式用到时必须填写，须在 0–90° 之间（不含 0 和 90）；「绳 2 与水平夹角 °」：所选方式用到时必须填写，须在 0–90° 之间（不含 0 和 90）；「斜面角 °」：所选方式用到时必须填写，须在 0–90° 之间（可为 0，不含 90）；「摩擦系数（可选）」：可留空（留空按 0），不能为负；「质量 2 kg」：所选方式用到时必须填写，须大于 0；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Tension Calculator (Vertical, Two Angled Ropes, Incline, Pulley With Two Masses) checks its inputs — "Mass kg": required, must be greater than 0; "Accel (up +)": optional (blank counts as 0); "g": required when the chosen option uses it; "Rope 1 angle from horizontal °": required when the chosen option uses it, must be between 0 and 90° (not 0 or 90); "Rope 2 angle from horizontal °": required when the chosen option uses it, must be between 0 and 90° (not 0 or 90); "Incline angle °": required when the chosen option uses it, must be from 0 up to but not including 90°; "Friction coefficient (optional)": optional (blank counts as 0), cannot be negative; "Mass 2 kg": required when the chosen option uses it, must be greater than 0; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.

有哪些选项可以调整？What options can I adjust?

「张力计算器（竖直加速、斜拉两绳、斜面、滑轮两物体）」的设置：「情形」可选 竖直吊挂、两根斜绳、斜面、滑轮两物体（默认 竖直吊挂）；「质量 kg」默认 10；「加速度（向上为正）」默认 0；「g」默认 9.81；「绳 1 与水平夹角 °」默认 30；「绳 2 与水平夹角 °」默认 45；「斜面角 °」默认 30；「摩擦系数（可选）」默认 0；「质量 2 kg」默认 6。每次生成结果都按当前设置处理。Settings in the Tension Calculator (Vertical, Two Angled Ropes, Incline, Pulley With Two Masses) — "Set-up": Hanging vertically, Two angled ropes, Incline, Pulley with two masses (default Hanging vertically); "Mass kg" default 10; "Accel (up +)" default 0; "g" default 9.81; "Rope 1 angle from horizontal °" default 30; "Rope 2 angle from horizontal °" default 45; "Incline angle °" default 30; "Friction coefficient (optional)" default 0; "Mass 2 kg" default 6. The result always uses the current settings.

计算结果可以用于作业或工程吗？Can I use the results for homework or engineering?

「张力计算器（竖直加速、斜拉两绳、斜面、滑轮两物体）」使用教科书标准公式（公式写在说明和常见问题中），适合学习和估算核对；涉及安全的工程设计请以规范和实测为准。The Tension Calculator (Vertical, Two Angled Ropes, Incline, Pulley With Two Masses) uses standard textbook formulas (given in the description and FAQ), which is good for learning and checking estimates; for safety-critical engineering follow codes and measured values.

「张力计算器（竖直加速、斜拉两绳、斜面、滑轮两物体）」免费吗？需要注册吗？Is the Tension Calculator (Vertical, Two Angled Ropes, Incline, Pulley With Two Masses) free? Do I need an account?

「张力计算器（竖直加速、斜拉两绳、斜面、滑轮两物体）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Tension Calculator (Vertical, Two Angled Ropes, Incline, Pulley With Two Masses) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.