Introduction to Hooke's Law
Welcome to Hooke's Law for CCEA Further Mathematics (Unit AS 2: Mechanics 1). If you have ever stretched an elastic band, bounced on a trampoline, or watched a bungee jumper, you have seen elastic forces in action.
In this chapter, we explore what happens when strings and springs stretch and compress, how much force they exert, and how much energy they store. Don't worry if mechanics has felt intimidating before — we will break down every single formula step-by-step so you feel fully confident tackling exam questions.
Key Takeaway: Hooke's Law connects the force in a stretched or compressed object to how far it has changed from its original, unstretched shape.
1. Fundamental Definitions & Core Concepts
Before jumping into the mathematics, let's understand the core physical quantities involved in elastic systems.
A. Natural Length (\(l\) or \(l_0\))
The natural length, denoted by \(l\), is the original length of an elastic string or spring when no external forces act upon it (unstretched and uncompressed). In all calculations, natural length must be measured in metres (\(\text{m}\)).
B. Extension and Compression (\(x\))
When a force acts on an elastic body, its length changes:
• Extension (\(x\)): If the stretched length is \(L\), the extension is \(x = L - l\).
• Compression (\(x\)): If a spring is squeezed to a compressed length \(L\), the compression is \(x = l - L\).
Memory Tip: Always remember that \(x\) represents the change in length, never the total length itself!
C. Modulus of Elasticity (\(\lambda\))
The modulus of elasticity, denoted by the Greek letter lambda (\(\lambda\)), is a measure of the stiffness and material strength of the string or spring. It is measured in Newtons (\(\text{N}\)).
Did you know? The modulus of elasticity \(\lambda\) has a neat physical meaning: it is the theoretical tension required to double the string's natural length (i.e. when extension \(x = l\)).
D. Spring Constant / Stiffness (\(k\))
Sometimes problems refer to the spring constant or stiffness (\(k\)), measured in \(\text{N}\cdot\text{m}^{-1}\) (or \(\text{N}/\text{m}\)). The relationship between \(\lambda\) and \(k\) is:
\(k = \frac{\lambda}{l}\)
Key Takeaway: Natural length \(l\) is where extension is zero (\(x = 0\)). The force depends directly on extension \(x\), scaled by \(\lambda\) and \(l\) (or by \(k\)).
2. Hooke's Law Formula
Hooke's Law states that the tension or thrust force \(T\) in an elastic string or spring is directly proportional to its extension or compression \(x\), provided the elastic limit is not exceeded.
Mathematically, we write:
\(T = \frac{\lambda x}{l} = kx\)
Where:
• \(T\) = Tension (pulling inwards) or Thrust (pushing outwards) in Newtons (\(\text{N}\))
• \(\lambda\) = Modulus of elasticity in Newtons (\(\text{N}\))
• \(l\) = Natural length in metres (\(\text{m}\))
• \(x\) = Extension or compression in metres (\(\text{m}\))
• \(k\) = Spring constant in Newtons per metre (\(\text{N}\cdot\text{m}^{-1}\))
Modelling Assumptions to Keep in Mind:
1. Light: The string or spring has negligible mass. It has no kinetic energy or gravitational potential energy of its own, and tension is uniform throughout.
2. Elastic String vs. Elastic Spring:
• An elastic string can only be in tension (\(x > 0\)). When compressed or when its length \(L \le l\), the string goes slack, meaning \(T = 0\).
• An elastic spring can be in tension (when extended, \(L > l\)) AND in thrust/compression (when pushed shorter than natural length, \(L < l\)).
3. Standard Gravity: In CCEA Mechanics, standard acceleration due to gravity is \(g = 9.8\text{ ms}^{-2}\).
Key Takeaway: Strings go slack when \(L \le l\) (\(T = 0\)), but springs push back with thrust when compressed.
3. Elastic Potential Energy (EPE)
When you pull on an elastic string or compress a spring, you do work against the elastic force. This work is stored as Elastic Potential Energy (EPE).
Derivation via Integration:
Because the tension increases as extension increases, we find the work done by integrating the force over the distance stretched:
\(\text{EPE} = \int_{0}^{x} T \, du = \int_{0}^{x} \frac{\lambda u}{l} \, du = \left[ \frac{\lambda u^2}{2l} \right]_{0}^{x} = \frac{\lambda x^2}{2l}\)
Using the spring constant \(k = \frac{\lambda}{l}\), this can also be written as:
\(\text{EPE} = \frac{1}{2}kx^2\)
Important Energy Reminders:
• EPE is always a scalar quantity and is measured in Joules (\(\text{J}\)).
• Because \(x\) is squared (\(x^2\)), EPE is always positive or zero (\(\text{EPE} \ge 0\)).
• For an elastic string, \(\text{EPE} = 0\) whenever the string is slack (\(x = 0\)).
• For an elastic spring, \(\text{EPE} = \frac{\lambda x^2}{2l}\) when stretched by \(x\) AND when compressed by \(x\).
Key Takeaway: Work done stretching/compressing = \(\text{EPE} = \frac{\lambda x^2}{2l} = \frac{1}{2}kx^2\).
4. The Work-Energy Principle with Hooke's Law
Many CCEA exam questions ask you to find the speed of a particle, the maximum extension, or the height reached. These are best solved using the Work-Energy Principle.
The general conservation of energy equation is:
\(E_{\text{initial}} + W_{\text{external}} = E_{\text{final}} + W_{\text{loss}}\)
Where the total mechanical energy at any point consists of:
• Kinetic Energy: \(\text{KE} = \frac{1}{2}mv^2\)
• Gravitational Potential Energy: \(\text{GPE} = mgh\) (measured relative to a chosen reference datum level)
• Elastic Potential Energy: \(\text{EPE} = \frac{\lambda x^2}{2l}\)
Step-by-Step Method for Energy Problems:
Step 1: Choose a clear Datum Level: Pick a fixed horizontal level where \(\text{GPE} = 0\) (often the lowest point in the motion, or the point of release).
Step 2: Identify Positions: Clearly label Initial Position (1) and Final Position (2).
Step 3: Calculate Energies at Position 1: Write down \(\text{KE}_1\), \(\text{GPE}_1\), and \(\text{EPE}_1\).
Step 4: Calculate Energies at Position 2: Write down \(\text{KE}_2\), \(\text{GPE}_2\), and \(\text{EPE}_2\).
Step 5: Form the Equation: If no non-conservative external forces (like friction or air resistance) do work, set \(\text{Total Energy}_1 = \text{Total Energy}_2\):
\(\text{KE}_1 + \text{GPE}_1 + \text{EPE}_1 = \text{KE}_2 + \text{GPE}_2 + \text{EPE}_2\)
Key Takeaway: Energy cannot be created or destroyed. Setting up an energy balance table with KE, GPE, and EPE makes complex problems simple.
5. Static Equilibrium & Vertical Motion
When a particle hangs in equilibrium attached to an elastic string or spring, the upward tension balances the downward weight.
Equilibrium Condition:
\(T_0 = mg \implies \frac{\lambda x_0}{l} = mg\)
Rearranging gives the equilibrium extension \(x_0\):
\(x_0 = \frac{mgl}{\lambda}\)
Crucial Distinction: Equilibrium vs. Maximum Extension
A classic exam pitfall occurs when a mass is released from rest at the natural length of a vertical string:
• At the equilibrium position (\(x = x_0\)), the net force is zero, but the mass has gained kinetic energy and is travelling at its maximum speed.
• At the lowest point / maximum extension (\(x = x_{\text{max}}\)), the particle is momentarily at rest (\(v = 0\)). By conservation of energy:
\(mg x_{\text{max}} = \frac{\lambda x_{\text{max}}^2}{2l} \implies x_{\text{max}} = \frac{2mgl}{\lambda} = 2x_0\)
The maximum extension is twice the equilibrium extension!
Key Takeaway: At equilibrium, acceleration is zero and speed is maximum. At maximum extension, speed is zero and acceleration is directed upwards.
6. Common Pitfalls & Examiner Warnings
Make sure you avoid these common traps highlighted in examiner reports:
• Trap 1: Confusing Total Length with Extension: Never substitute the stretched length \(L\) into \(\frac{\lambda x}{l}\). Always calculate \(x = L - l\) first.
• Trap 2: Ignoring Slack Strings: Remember that an elastic string cannot push. If a particle on an elastic string travels above the natural length, \(T = 0\) and \(\text{EPE} = 0\); it moves freely under gravity alone.
• Trap 3: Unit Mismatches: Extensions and lengths are often stated in \(\text{cm}\). Always convert to metres (\(\text{m}\)) before substituting into Hooke's Law or EPE formulas (e.g. \(20\text{ cm} = 0.2\text{ m}\)).
• Trap 4: Modulus (\(\lambda\)) vs. Spring Constant (\(k\)): Pay close attention to which one is given. If using \(\lambda\), include the natural length \(l\) in the denominator (\(\frac{\lambda x^2}{2l}\)). If using \(k\), do not divide by \(l\) again (\(\frac{1}{2}kx^2\)).
• Trap 5: Inconsistent GPE Datum: Stick strictly to one reference level throughout your entire energy equation. If an object falls a distance \(h\) below your datum, its GPE is \(-mgh\).
7. Quick Review Summary
1. Hooke's Law Formula: \(T = \frac{\lambda x}{l} = kx\)
2. Elastic Potential Energy: \(\text{EPE} = \frac{\lambda x^2}{2l} = \frac{1}{2}kx^2\)
3. Spring Constant Relation: \(k = \frac{\lambda}{l}\)
4. Equilibrium Condition (Vertical): \(T = mg \implies \frac{\lambda x_0}{l} = mg\)
5. Total Mechanical Energy: \(E_{\text{total}} = \frac{1}{2}mv^2 + mgh + \frac{\lambda x^2}{2l}\)
6. CCEA Gravity Constant: \(g = 9.8\text{ ms}^{-2}\)