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 $x$ and $y$
  - or, in the case of equilibrium extensions, $e_{A}$ and $e_{B}$
You will need an extra equation relating $x$ to $y$ , found by summing all the lengths
- e.g. if the total distance is $5 l$ then $l + x + y + l = 5 l$
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 $x$ or $y$ 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 $T = \frac{λ}{l} x$ where $x$ 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 $T = \frac{λ}{(\frac{l}{2})} (\frac{x}{2})$
  - This simplifies to $T = \frac{λ}{l} x$ (showing that both methods give the same answer)

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

A particle of mass $m$ kg has a weight of $m g$ 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 $T_{1}$ and $T_{2}$
If angles are not given, use the geometry of the situation to find $\sin θ$ and $\cos θ$ (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

Exam Tip

Using your own subscripts can help to avoid confusing tensions and extensions from situations with multiple strings or springs

Worked example

A light spring of natural length $2 l$ metres and modulus of elasticity $2 m g$ N has one end attached to the point A on a ceiling and its other end attached to a particle of mass $m$ kg suspended vertically beneath A.

A different spring of natural length $l$ metres and modulus of elasticity $4 m g$ N has one end attached to the particle and its other end attached to the point B on the floor, where B is a distance of $5 l$ metres vertically beneath A. The system is in equilibrium.

Find, in terms of $l$ , the height of the particle above the floor.

worked example for problem solving with springs and strings, with an algebra-based question (part 1)

part 2 of worked example for problem solving with springs and strings, with an algebra-based question

Problem Solving with Strings & Springs (Edexcel A Level Further Maths: Further Mechanics 1)

Revision Note