Elasticity, circular motion and variable dynamics · 弹性、圆周运动与变动力学
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| extension/ekˈstenʃn/ | 伸长量 | shēn cháng liàng |
Which length stretches the spring?
- A stretched spring stores energy. A particle moving in a circle needs a resultant force towards the centre, even at constant speed.
- This lesson studies extension 伸长量: The increase in an elastic string's or spring's length from its natural length.
哪一长度表示弹簧的伸长量？
- 拉伸的弹簧储存弹性势能。做圆周运动的质点需要指向圆心的合力，即使速率保持不变。
- 本课研究 extension 伸长量：弹性绳或弹簧的长度相对于其自然长度的增加量。
Choose the mathematical structure
- For an elastic string, tension=λx/l when stretched, with natural length l and modulus λ. Elastic energy=λx²/(2l). For circular motion, radial acceleration=v²/r; resolve actual forces in the radial direction.
- 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
- 对于弹性绳，拉伸时张力 T=λx/l，其中 l 为自然长度，λ 为劲度系数。弹性势能 E=λx²/(2l)。对于圆周运动，径向加速度 a=v²/r；需在径向方向上分解实际受力。
- 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 extension? · 哪个描述正确定义了伸长量？
The increase in an elastic string's or spring's length from its natural length. · 弹性绳或弹簧长度相对于其自然长度的增加量。
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.
With λ=40 N,l=2 m,x=0.3 m, tension=40×0.3/2=6 N and elastic energy=40×0.3²/(2×2)=0.9 J. A 2 kg particle at speed 3 m/s on radius 1.5 m needs resultant inward force=2×3²/1.5=12 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。
当 λ=40 N，l=2 m，x=0.3 m 时，张力 =40×0.3/2=6 N，弹性势能 =40×0.3²/(2×2)=0.9 J。质量为 2 kg 的质点以 3 m/s 的速率在半径为 1.5 m 的圆周上运动，所需向内的合力 =2×3²/1.5=12 N。
Find tension for λ=40,l=2,x=0.3. · 求 λ=40, l=2, x=0.3 时的张力。
T=λx/l=40×0.3/2=6 N.
Test a tempting shortcut
- Centripetal force is the name of the inward resultant, not an extra force to add to the diagram. A slack string has no tension. Distinguish extension from total stretched length.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Centripetal force is an additional force that must be added to all the real forces. This claim is false. Explain which definition or assumption it violates.
检验一个诱人的捷径
- 向心力是指向圆心的合力的名称，并非需要添加到受力图上的额外力。松弛的绳子没有张力。需区分“伸长量”与“总拉伸长度”。
- 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。
向心力是一种必须添加到所有真实受力之上的额外力。此说法是错误的。请解释它违反了哪一定义或假设。
Find elastic energy for those values. · 求这些数值对应的弹性势能。
Elastic energy=λx²/(2l)=40×0.09/4=0.9 J. · 弹性势能=λx²/(2l)=40×0.09/4=0.9 J。
Centripetal force is an additional force that must be added to all the real forces. · 向心力是一种必须添加到所有真实力之上的附加力。
Centripetal force is the name of the inward resultant, not an extra force to add to the diagram. A slack string has no tension. Distinguish extension from total stretched length. · 向心力是向内合力的名称，而不是需要加到受力图上的额外力。松弛的绳子没有张力。注意区分伸长量与总拉伸长度。
Interpret a new situation
- For variable acceleration, use a=dv/dt or a=v dv/dx when appropriate. Combine the chosen motion relation with the force or energy model, then check the permitted range of motion.
- 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
- 对于变加速运动，根据情况使用 a=dv/dt 或 a=v dv/dx。将所选的运动学关系与力学或能量模型结合，然后检查允许的运动范围。
- 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。
Find inward resultant for m=2,v=3,r=1.5. · 求 m=2, v=3, r=1.5 时的向内合力。
Inward resultant=mv²/r=2×9/1.5=12 N. · 向内合力=mv²/r=2×9/1.5=12 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 M3. 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.
The increase in an elastic string's or spring's length from its natural length. Choose the relationship, show the method, check its assumptions and interpret the result.
将其应用于你的课程
- edexcel IAL 高等数学；官方单元 M3。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。
弹性绳或弹簧的长度相对于其自然长度的增加量。选择正确的关系式，展示推导方法，检验其适用假设并解读结果。