Elasticity, circular motion and variable dynamics
| English | 中文 | Pinyin |
|---|---|---|
| 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.
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.
Which description correctly defines extension?
The increase in an elastic string's or spring's length from its natural length.
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.
Find tension for λ=40,l=2,x=0.3.
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.
Find elastic energy for those values.
Elastic energy=λx²/(2l)=40×0.09/4=0.9 J.
Centripetal force is an additional force that must be added to all the real forces.
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.
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.
Find inward resultant for m=2,v=3,r=1.5.
Inward resultant=mv²/r=2×9/1.5=12 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.
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.