Moments, equilibrium and centres of mass · 力矩、平衡与重心
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| moment/ˈməʊmənt/ | 力矩 | lì jǔ |
Why does a longer spanner help?
- A long spanner turns a nut more easily. The turning effect depends on perpendicular distance as well as force.
- This lesson studies moment 力矩: Force multiplied by the perpendicular distance from its line of action to a pivot.
为什么更长的扳手更有用？
- 长扳手更容易拧动螺母。转动效果既取决于力的大小，也取决于垂直距离。
- 本课研究力矩 (moment)：力乘以其作用线到支点的垂直距离。
Choose the mathematical structure
- For equilibrium, both the resultant force and resultant moment are zero. Take moments about a useful pivot to remove unknown forces through that point. A combined centre of mass is a mass-weighted mean position.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
选择数学结构
- 对于平衡状态，合力与合力矩均为零。对有用的支点取力矩可消除通过该点的未知力。组合质心是质量加权平均位置。
- 计算前请先明确允许的输入项和单位。方程应表达关系本身，而不仅仅是记录计算器按键过程。
Which description correctly defines moment? · 哪项描述正确定义了力矩？
Force multiplied by the perpendicular distance from its line of action to a pivot. · 力与其作用线到支点的垂直距离之积。
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
A uniform 4 m beam weighs 100 N and is supported at both ends. An extra 60 N acts 1 m from the left. Moments about the left give 4R_right=100×2+60×1=260, so R_right=65 N. Vertical equilibrium gives R_left=95 N.
通过验证案例进行推导
- 将结果与初始量进行比对。代入原始关系式，或在适当情况下对比图表与数值答案。
一根均匀的4 m长梁重100 N，两端由支架支撑。一个额外的60 N的力作用在距左端1 m处。对左端取力矩得4R_right=100×2+60×1=260，因此R_right=65 N。竖直方向平衡给出R_left=95 N。
Moments, equilibrium and centres of mass · 力矩、平衡与重心
For equilibrium, both the resultant force and resultant moment are zero · 处于平衡状态时，合力和合力矩均为零。
Distinguish the force from its moment about a chosen pivot. · 区分所选支点处的力与其力矩。
Find the moment of 20 N with perpendicular arm 0.3 m. · 求大小为 20 N、垂直力臂为 0.3 m 的力矩。
Moment=force×perpendicular arm=20×0.3=6 N m. · 力矩=力×垂直力臂=20×0.3=6 N m。
Test a tempting shortcut
- Use the perpendicular lever arm, not a diagonal distance to the point of application. A zero resultant force alone does not prevent rotation. A support that would need a negative normal reaction may lose contact.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Zero resultant force by itself is enough for a rigid body's equilibrium. This claim is false. Explain which definition or assumption it violates.
检验一个诱人的捷径
- 使用垂直力臂，而不是指向作用点的对角距离。仅合力为零并不能防止旋转。如果支撑面需要产生负法向反作用力，则可能失去接触。
- 当捷径失效时，找出其违背的假设。保留精确值直到题目要求的最终舍入步骤。
仅合力为零就足以保证刚体的平衡。此说法错误。请解释该主张违反了哪个定义或假设。
For the beam in the example, find the right reaction. · 对于示例中的梁，求右端支反力。
Taking moments about the left gives 4R=100×2+60×1, so R=65 N. · 对左端取矩得 4R=100×2+60×1，故 R=65 N。
Zero resultant force by itself is enough for a rigid body's equilibrium. · 仅合力为零并不足以保证刚体处于平衡状态。
Use the perpendicular lever arm, not a diagonal distance to the point of application. A zero resultant force alone does not prevent rotation. A support that would need a negative normal reaction may lose contact. · 应使用垂直力臂，而非指向作用点的斜向距离。仅合力为零无法阻止转动。若支撑物所需的法向反作用力为负值，则可能脱离接触。
Interpret a new situation
- For a lamina made from parts, choose a common origin and tabulate each signed mass-area contribution. A removed region has negative mass in that calculation, not a new physical negative material.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
解读新情境
- 对于由多部分组成的薄板，请选择一个共同的原点，并以表格形式列出每个带符号的质量面积贡献。被移除的区域在该计算中具有负质量，而非新的物理负物质。
- 完整解答需给出数学结果并解释其含义。检查其在所述背景下是否可行。
For that beam, find the left reaction. · 对于该梁，求左端支反力。
Vertical forces balance: L=100+60-65=95 N. · 垂直方向受力平衡：L=100+60-65=95 N。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- edexcel IAL further mathematics; official unit M1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
Force multiplied by the perpendicular distance from its line of action to a pivot. Choose the relationship, show the method, check its assumptions and interpret the result.
将其应用于你的课程
- edexcel IAL 进阶数学；官方单元 M1。其他单元的拓展内容已在范围审查中列出，不属于额外的加分要求。
- 在给出最终答案前先展示解题方法，并遵循试卷的计算器及公式使用规则。通过定位第一个无效步骤来复盘错误答案。
力乘以其作用线到支点的垂直距离。选择关系式，展示方法，检查其假设并解释结果。