Inclined planes, friction and connected particles · 斜面、摩擦力和连接体问题
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| limiting friction/ˈlɪmɪtɪŋ ˈfrɪkʃn/ | 极限摩擦力 | jí xiàn mó cā lì |
Is friction already at its maximum?
- A crate on a slope is at rest. Friction can adjust to balance the force down the slope; it need not already be at its maximum.
- This lesson studies limiting friction 极限摩擦力: The maximum static friction before slipping, equal to μ times the normal reaction in the model.
Choose the mathematical structure
- Resolve parallel and perpendicular to the plane. Weight components are mg sinθ and mg cosθ. Static friction satisfies F≤μR and equals μR only at the limiting case. Write separate equations for connected particles.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines limiting friction? · 以下哪项描述正确定义了极限摩擦力？
The maximum static friction before slipping, equal to μ times the normal reaction in the model. · 滑动前的最大静摩擦力，等于模型中μ乘以法向反力。
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 5 kg crate on a 30° slope has R=5×9.8×cos30°≈42.44 N. The downhill weight component is 24.5 N. With μ=0.6, maximum friction≈25.46 N, so equilibrium is possible with friction 24.5 N.
Inclined planes, friction and connected particles · 斜面、摩擦力和连接体问题
Resolve parallel and perpendicular to the plane · 沿平行于斜面和垂直于斜面的方向分解力
Compare the model with the worked case and explain one change. · 对比模型与已解案例，并说明其中一处变化。
Find the downhill component of weight for m=5 kg,θ=30°,g=9.8. · 当 m = 5 kg，θ = 30°，g = 9.8 时，求重力的下滑分力。
Resolve weight along the slope: 5×9.8×sin30°=24.5 N. · 沿斜面分解重力：5×9.8×sin30°=24.5 N。
Test a tempting shortcut
- Friction opposes motion or the tendency to move, not always the coordinate direction. A taut light inextensible string over a smooth pulley gives equal tension and a common acceleration magnitude; each assumption has a job.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Static friction is always exactly μR even when the body is not about to slip. This claim is false. Explain which definition or assumption it violates.
Find limiting friction when μ=0.6 and R=40 N. · 当μ=0.6且R=40 N时，求极限摩擦力。
Limiting friction=μR=0.6×40=24 N. · 极限摩擦力=μR=0.6×40=24 N。
Static friction is always exactly μR even when the body is not about to slip. · 静摩擦力并非总是等于μR，即使物体尚未发生滑动。
Friction opposes motion or the tendency to move, not always the coordinate direction. A taut light inextensible string over a smooth pulley gives equal tension and a common acceleration magnitude; each assumption has a job. · 摩擦力阻碍运动或运动趋势，而不仅仅是坐标方向。跨过光滑定滑轮的轻绳不可伸长，张力处处相等，各部分具有相同大小的加速度；每一项假设都有其特定作用。
Interpret a new situation
- Start with force diagrams and a proposed direction of motion. If the calculated direction conflicts with the friction assumption, revisit the model instead of keeping inconsistent signs.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
If a 2 kg particle has net force 7 N, find its acceleration. · 若一个质量为2 kg的质点受到的合力为7 N，求其加速度。
Use resultant force: a=7/2=3.5 m/s². · 使用合力公式：a=7/2=3.5 m/s²。
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.
The maximum static friction before slipping, equal to μ times the normal reaction in the model. Choose the relationship, show the method, check its assumptions and interpret the result.