Problem Solving with Strings & Springs (Edexcel A Level Further Maths): Revision Note

Problem solving with strings & springs

What if the particle is fixed between two different springs?

• There will be two different tensions and extensions

  • For extensions, you can use and 

    • or, in the case of equilibrium extensions, and 

• You will need an extra equation relating to , found by summing all the lengths

  • e.g. if the total distance is  then

• Springs can go into compression, but if you're not told (or it's not clear), draw both springs under tension

  • If it turns out that or  are negative later on, then you know they were actually under compression

• If a particle is attached to the midpoint of a spring, you can either

  • Treat the spring as a whole 

    • The two tensions either side have the form where  is the total extension

  • Or treat the particle as being fixed between two half-springs, halving their lengths (natural lengths and extensions), but keeping the same modulus of elasticity (the material they're made from hasn't changed)

    • The two tensions either side have the form

    • This simplifies to  (showing that both methods give the same answer)

What if the particle is suspended at angles by two elastic strings?

• A particle of mass  kg has a weight of  N acting downwards

• The particle could be held up (suspended) by two light elastic strings acting at angles to the horizontal

  • Label the different tensions  and 

• If angles are not given, use the geometry of the situation to find and  (by trigonometry)

  • This helps later for resolving the tensions in Newton's 2nd Law

• If a particle is attached to the midpoint of an elastic string and forms a triangle due to its weight, you can either

  • Treat the string as a whole (calculations with full extension and full natural length)

  • Or treat the setup as a particle attached to two identical half-strings (each with half the natural length, half the extension, but the same modulus of elasticity)

  • In either case, this situation will be symmetric

    • If the particle is pulled vertically downwards and released, it will accelerate vertically upwards

What other questions can be asked about springs and strings?

• There are so many different situations that it's impossible to know, but the tools used are the same

  • Newton's 2nd law, Hooke's law and the work-energy principle

• If the situation has uniform rods suspended by light elastic strings, you may also need to take moments

• Some questions may be more algebraic

• Other questions could have a change halfway through

  • e.g. the string breaks and the particle becomes a projectile