Elastic strings and springs - Further Mathematics - Pearson A-Level

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Welcome to Elastic Strings and Springs!

Hello future Further Mathematician! This chapter, part of Unit M3 (Mechanics 3), takes the dynamics and energy concepts you mastered in M1 and M2 and applies them to objects that stretch and recoil: elastic strings and springs.

Don't worry if the formulas look complicated at first! We'll break down the core principle—Hooke's Law—and show you how incredibly useful it is for solving real-world problems involving potential energy and motion. Mastering this chapter is essential preparation for studying Simple Harmonic Motion (SHM) later.

Section 1: Hooke's Law – The Foundation of Elasticity

When an elastic string or spring is stretched or compressed, it resists that change. This resistance is called Tension (T) for strings, or sometimes Thrust/Force (F) for springs, and it tries to restore the object to its original length. The relationship governing this restorative force is Hooke's Law.

What is Hooke's Law?

Hooke's Law states that, provided the elastic limit of the material is not exceeded, the tension (or restoring force) in the object is directly proportional to the extension or compression.

In plain English: The more you pull it, the harder it pulls back!

Key Definitions

Natural Length (l): This is the length of the string or spring when no forces are acting on it (i.e., when it is completely relaxed).
Extension or Compression (x): This is the change in length from the natural length. If the current length is $L$, then $x = |L - l|$.
Tension (T): The pulling force within the string or spring.

The Core Formula

We convert the proportionality $T \propto x$ into an equation by introducing a constant related to the material's stiffness. This leads to the fundamental formula for M3 elasticity:

$$T = \frac{\lambda x}{l}$$

Let's define the new symbol:

$ \lambda $ – The Modulus of Elasticity

This constant, $\lambda$, is a measure of how stiff the material is. It is the force required to double the natural length of the string or spring.

Units of $\lambda$ are usually Newtons (N).
A large $\lambda$ means the object is stiff (hard to stretch).
A small $\lambda$ means the object is soft (easy to stretch).

💡 Memory Aid for Hooke's Law:

Think of Tension being determined by the Length ($l$) and the Lambda ($\lambda$) and the eXtension ($x$). It’s Time for Lambda X over L.

Crucial Distinction: Strings vs. Springs

This is a vital concept in M3 problems, especially when things become compressed:

1. Elastic Strings:
Strings can only exert a force (Tension) when they are stretched ($x > 0$).
If the string is compressed or returns to its natural length ($x \le 0$), it goes slack.
The Tension is zero when the string is slack: $T = 0$.

2. Elastic Springs:
Springs can exert a force whether they are stretched (Tension) or compressed (Thrust/Compression).
The Hooke's Law formula $T = \frac{\lambda x}{l}$ applies to both stretching and compressing.
If $x$ represents extension, a negative $x$ represents compression, and the resulting negative $T$ indicates a force acting in the opposite direction (a push, or thrust).

Key Takeaway (Section 1):

Hooke's Law gives us the force required to maintain a certain extension $x$. Always remember to check if a string goes slack ($T=0$)!

Section 2: Elastic Potential Energy (EPE)

When you stretch a string or spring, you are doing work against the tension force. Because the system is elastic, this work is not lost; it is stored within the material as Elastic Potential Energy (EPE), ready to be converted back into kinetic energy or gravitational potential energy.

Analogy Corner:

Think of a catapult. The energy you use to pull the elastic band back (work done) is stored as EPE. When you release it, the EPE is converted instantly into the Kinetic Energy of the projectile.

Calculating Elastic Potential Energy

Since the Tension $T$ increases linearly with the extension $x$, we cannot use the simple "Work = Force $\times$ Distance" formula. Instead, we must use integration (or the area under the T-x graph, which is a triangle).

The total work done in stretching an object from zero extension to extension $x$ is stored as EPE. This is calculated as:

$$EPE = \frac{1}{2} \times \text{Force} \times \text{Extension} = \frac{1}{2} T x$$

Substituting Hooke's Law $T = \frac{\lambda x}{l}$ into this equation gives us the definitive formula for EPE:

$$EPE = \frac{\lambda x^2}{2l}$$

Notice the $x^2$! This is very similar to the $\frac{1}{2}mv^2$ (KE) and the $\frac{1}{2} k x^2$ used in SHM later. Energy is always proportional to the square of velocity or displacement in these systems.

Work Done: An Alternative View

The work done by the tension force when the object relaxes is equal to the EPE released. The work done against the tension force when stretching is the EPE stored.

$$W = \Delta EPE$$

Common Mistakes to Avoid!

The two biggest errors when using EPE:

Forgetting the $x^2$: The formula is $EPE = \frac{\lambda x^2}{2l}$, not $\frac{\lambda x}{2l}$.
Mixing up $l$ and $x$: Remember that $l$ is the natural length (constant), and $x$ is the extension (variable).

Key Takeaway (Section 2):

Elastic Potential Energy $EPE = \frac{\lambda x^2}{2l}$ represents the energy stored due to stretching or compression. Use it in energy conservation equations just like KE and GPE.

Section 3: Applications in Dynamics and Energy Conservation

In M3, we combine elasticity with the other forces you already know: Gravity (GPE), Kinetic Energy (KE), and sometimes air resistance (though usually negligible).

Application A: Conservation of Mechanical Energy (CoE)

This is the most common way to solve elasticity problems. If there are no non-conservative forces (like air resistance or external friction), the total mechanical energy of the system remains constant.

$$\text{Initial Energy} = \text{Final Energy}$$

$$KE_i + GPE_i + EPE_i = KE_f + GPE_f + EPE_f$$

Step-by-Step for CoE Problems:

Identify the States: Choose two points in time/position (e.g., initial release point and maximum speed point, or initial release point and lowest point reached).
Define the Datum: Choose a reference level for GPE ($GPE = mgh$). This is crucial! Often, the lowest point reached is the easiest datum ($h=0$).
Calculate Extensions (x): Determine the extension $x$ at both the initial and final states. Remember that $x$ is measured from the natural length $l$.
Substitute and Solve: Plug all six terms into the conservation equation and solve for the unknown velocity or displacement.

Example Context: A particle of mass $m$ is attached to a vertical spring of natural length $l$. It is released from rest at the level of the spring’s attachment point.

State 1 (Top, release point): $KE_i = 0$, $EPE_i = 0$ (string/spring is unstretched).

State 2 (Bottom, lowest point, distance $D$ below datum): $KE_f = 0$, $GPE_f = -mgD$, $EPE_f = \frac{\lambda D^2}{2l}$.

Equation: $0 + 0 + 0 = 0 - mgD + \frac{\lambda D^2}{2l}$. We can then solve for $D$.

Application B: Using F = ma (Dynamics)

When you need to find acceleration (a), you must revert to Newton’s Second Law. This is especially important when finding the point of maximum speed or the equilibrium position.

$$\text{Net Force} = ma$$

At any point, the forces acting on the particle are:

Weight ($mg$)
Tension ($T = \frac{\lambda x}{l}$)
Other forces (e.g., normal reaction, air resistance)

Finding the Equilibrium Position:
The equilibrium position is where the particle *could* rest. At this point, the acceleration is zero ($a=0$), meaning Net Force = 0.

Example: A particle hanging vertically: At equilibrium, the upward tension must exactly balance the weight.

$$\text{Upward Forces} = \text{Downward Forces}$$ $$T = mg$$ $$\frac{\lambda x}{l} = mg$$

You can then solve for $x$, the extension at the equilibrium position.

Did You Know?

The elastic properties of materials are crucial in engineering! Bridges, aircraft wings, and even simple shock absorbers rely on understanding Hooke’s Law and EPE to ensure safety and function correctly without failing (reaching the "elastic limit").

Quick Review: Formulas You Must Know

These two formulas are the heart of the chapter. Check your definitions of the variables every time!

Hooke's Law (Force):
$$T = \frac{\lambda x}{l}$$

Elastic Potential Energy (Stored Energy):
$$EPE = \frac{\lambda x^2}{2l}$$

Key Takeaway (Section 3):

Most M3 elasticity problems are solved using Conservation of Energy (integrating EPE, KE, and GPE). Use $F=ma$ when specifically asked to find acceleration or the equilibrium position.