Further Mathematics (9665) Study Notes: Elastic Strings and Springs (FM2.3)

Hello! Welcome to one of the most practical and interesting topics in Further Mechanics: Elastic Strings and Springs. This chapter is all about understanding how flexible materials store and release energy. Whether you are analyzing a bungee jump, a catapult, or just a simple spring scale, the principles here are fundamental. Don't worry if this seems tricky at first; we will break down the concepts, especially Hooke's Law and Elastic Potential Energy, into clear, manageable steps!

The key to success here is careful application of definitions and the Principle of Conservation of Energy.

1. Hooke's Law: The Foundation of Elasticity

Hooke's Law describes the relationship between the force applied to an elastic object (like a spring or string) and how much it stretches.

What is Hooke's Law?

In simple terms, for an elastic material, the tension (T) created is directly proportional to the extension (e) it experiences, provided we don't stretch it too far (staying within the elastic limit).

The standard formula used in Further Mathematics (9665) is:

$$\b{T = \frac{\lambda}{l}e}$$

Where:

  • \(T\) is the Tension (Force, measured in Newtons, N).
  • \(\lambda\) is the Modulus of Elasticity (N). This is a constant unique to the material itself.
  • \(l\) is the Natural Length (m). This is the length of the string or spring when no force is acting on it.
  • \(e\) is the Extension (m). This is the length stretched beyond the natural length.

Important Distinction: Strings vs. Springs

  • Elastic String: Can only generate Tension. It only obeys Hooke's Law if it is stretched (\(e > 0\)). If the length is less than or equal to the natural length (\(e \le 0\)), the tension \(T = 0\).
  • Elastic Spring: Can generate Tension (when stretched) and Thrust/Compression (when squashed). It obeys Hooke's Law for both extension and compression, meaning \(e\) can be positive or negative.
Key Takeaway: Hooke's Law

The force generated in an elastic object is proportional to how much it has stretched. Always check if you are dealing with a string (cannot push) or a spring (can push and pull).

2. Modulus of Elasticity (\(\lambda\)) and Stiffness (\(k\))

The relationship \(\b{T = \frac{\lambda}{l}e}\) can be simplified by defining a single constant that incorporates both the material property (\(\lambda\)) and its physical size (\(l\)).

2.1. The Stiffness Constant (\(k\))

We often define the Stiffness or Spring Constant, \(k\), as:

$$\b{k = \frac{\lambda}{l}}$$

Using this, Hooke's Law becomes the more compact form:

$$\b{T = ke}$$

(You may recognize this form from AS Physics, but in Further Maths, you must be comfortable using both \(k\) and \(\lambda\) forms.)

Analogy: Why do we have both \(\lambda\) and \(k\)?

Imagine you have a coil of wire (the material) and you cut off a piece to make a spring (the object).

  • \(\lambda\) (Modulus of Elasticity) describes the fundamental property of the wire itself (how intrinsically stretchy the metal is).
  • \(l\) (Natural Length) describes how much wire you used.
  • \(k\) (Stiffness) describes the property of the final spring. A shorter spring (smaller \(l\)) made from the same wire will feel much stiffer (larger \(k\)).

Struggling Students Tip: If a question changes the length of the string/spring (e.g., cutting it in half), the value of \(\lambda\) stays the same, but the value of \(k\) changes because \(l\) changes. If the question simply asks for the force, using \(T=ke\) is often easier, provided you know \(k\).

Quick Review: \(\lambda\) vs. \(k\)
  • \(k\) is the property of the specific object.
  • \(\lambda\) is the property of the material.
  • Relationship: \(\b{k = \lambda / l}\).

3. Work Done and Elastic Potential Energy (EPE)

Because the tension \(T\) is not constant—it increases as the object stretches—we cannot use the simple formula \(W = Force \times Distance\). Instead, we must use integration to find the work done in stretching the object.

3.1. Work Done by a Variable Force

The syllabus requires knowledge and use of the formula for work done by a variable force, \(F\), acting along a line of motion:

$$\b{W = \int F \, dx}$$

Where \(x\) is the displacement from the starting point.

In the case of an elastic string or spring, the work done in stretching it from zero extension to an extension \(e\) is the Elastic Potential Energy (EPE) stored in the object.

3.2. The Elastic Potential Energy Formula

Integrating \(W = \int_0^e T \, dx\) where \(T = kx\) (using \(x\) as the extension variable), we get the formula for EPE. Students are expected to quote this formula unless explicitly asked to derive it.

$$\b{EPE = \frac{1}{2}ke^2}$$

or, using the modulus of elasticity \(\lambda\):

$$\b{EPE = \frac{\lambda e^2}{2l}}$$

Note: The unit for EPE is Joules (J), as it represents stored energy.

Did you know?

EPE is the mechanical equivalent of charging a battery. When you stretch a spring, you are storing energy; when you release it (like letting go of a bowstring), that stored EPE converts into Kinetic Energy (KE).

Common Mistake to Avoid:

When calculating EPE, remember to use the extension \(e\), not the total length of the string. The extension is given by: \(e = \text{Current Length} - \text{Natural Length } l\).

Key Takeaway: Elastic Potential Energy

EPE is the energy stored due to stretching or compressing an elastic object, calculated using \(EPE = \frac{1}{2}ke^2\). This energy must be included in conservation of energy calculations.

4. Solving Problems using Conservation of Energy

Most complex problems involving elastic strings and springs require applying the Principle of Conservation of Mechanical Energy (PCME).

Step-by-Step Guide to Using PCME

The total energy in a closed system (where only conservative forces like gravity and tension/thrust are acting) remains constant:

$$\b{E_{Initial} = E_{Final}}$$

$$\b{(KE + GPE + EPE)_{Initial} = (KE + GPE + EPE)_{Final}}$$

Step 1: Define Your System and Reference Points

Choose a point where the object starts (Initial) and a point where you want to find information (Final). Crucially, decide on a zero level for Gravitational Potential Energy (GPE)—often the lowest point reached by the particle, or the starting point.

Step 2: Calculate Initial Energies

  • Initial KE: \(\frac{1}{2}mv^2\). (If released from rest, KE = 0).
  • Initial GPE: \(mgh\). (The height \(h\) relative to your chosen zero level).
  • Initial EPE: \(\frac{1}{2}ke^2\). (If the string is at its natural length, EPE = 0).

Step 3: Calculate Final Energies

Repeat Step 2 for the final position, paying close attention to the final velocity, height, and, most importantly, the final extension \(e\).

Step 4: Form and Solve the Equation

Set the totals equal and solve for the unknown variable (e.g., maximum extension, maximum speed, or maximum height).

Example Scenario: A particle of mass \(m\) is attached to an elastic string of natural length \(l\) and modulus \(\lambda\). It is released from rest at the point where the string is just taut (zero extension) and falls vertically.

  • Initial State (Just Taut, \(v=0\)): Set this height as GPE = 0. KE = 0. EPE = 0. Total Initial Energy = 0.
  • Final State (Maximum Extension \(e\)): The particle has fallen a total distance of \(l + e\).
    • Final KE: 0 (since velocity is zero at the lowest point).
    • Final GPE: \(-mg(l+e)\) (negative because it's below the starting zero level).
    • Final EPE: \(\frac{\lambda e^2}{2l}\).

By PCME: \(0 = 0 - mg(l+e) + \frac{\lambda e^2}{2l}\). You then solve this quadratic equation for \(e\).

Key Takeaway: Solving Problems

In FM2 mechanics problems involving elastic strings/springs, the most powerful tool is the Conservation of Energy. Systematically calculate KE, GPE, and EPE at your start and end points.

Good luck! By mastering Hooke's Law and EPE, you have unlocked the crucial tools needed for tackling oscillation problems later in the syllabus.