Moments, equilibrium and centres of mass
| English | 中文 | Pinyin |
|---|---|---|
| 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.
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.
Which description correctly defines moment?
Force multiplied by the perpendicular distance from its line of action to a pivot.
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.
Moments, equilibrium and centres of mass
For equilibrium, both the resultant force and resultant moment are zero
Distinguish the force from its moment about a chosen pivot.
Find the moment of 20 N with perpendicular arm 0.3 m.
Moment=force×perpendicular arm=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.
For the beam in the example, find the right reaction.
Taking moments about the left gives 4R=100×2+60×1, so R=65 N.
Zero resultant force by itself is enough for a rigid body's equilibrium.
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.
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.
For that beam, find the left reaction.
Vertical forces balance: L=100+60-65=95 N.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
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.
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.