Work Done by a Variable Force Revision Notes for Leaving Cert Applied Maths

Work Done by a Variable Force

Understanding the concept

When we studied constant forces, we learned that work done is simply calculated using a straightforward formula. However, in many real-world situations, the force applied changes as we move through the displacement. This requires us to use calculus to find the total work done.

For a constant force, work is calculated using the familiar equation $W = Fd$ . But when dealing with a variable force, we need a more sophisticated approach using integration.

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The transition from constant to variable forces represents a significant step in physics problem-solving. While constant force problems can be solved with basic algebra, variable force problems require the power of calculus to account for continuously changing conditions.

The variable force formula

When force varies with position, the simple $W = Fd$ formula no longer applies. Instead, we must use integration to calculate work done across small increments of displacement.

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The fundamental formula for work done by a variable force is:

$W = \int\_A^B F(x)dx$

Where:

$F(x)$ represents the force as a function of distance $x$ from some fixed reference point
The integration limits are between two positions A and B
$x$ represents the distance variable

This formula allows us to calculate the total work done when the force changes continuously throughout the motion by summing up infinitesimally small amounts of work done over each tiny displacement.

Worked example with elastic strings

Let's examine a practical problem involving an elastic string to understand how this works in practice. Elastic strings provide an excellent example of variable forces because they follow Hooke's Law, where force increases linearly with extension.

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Worked Example: Elastic String Stretching

Problem setup: An elastic string has an elastic constant of 14 N/m and a natural length of 50 cm. We need to find work done in different stretching scenarios.

Method comparison

There are two ways to approach this type of problem, depending on how we define our position variable:

Method 1 - Define x as total string length

Use Hooke's Law: $F = k(l - l\_0) = 14(x - 0.5) = 14x - 7$
This accounts for the natural length in the force equation

Method 2 - Define x as extension only

Use Hooke's Law: $F = kx = 14x$
This treats $x$ as the stretch beyond natural length

Both methods give identical results when applied correctly.

Key calculations

Part (i): Stretching from natural length (50 cm) to 1.5 m

Extension changes from 0 to 1 m
Work done = 7J (using either method)

Part (ii): Further stretching from 1.5 m to 2.5 m

Extension changes from 1 m to 2 m
Additional work done = 21J

Part (iii): Total potential energy when string is 2.5 m long

Extension is 2 m from natural length
Potential energy = 28J (equal to total work done from natural length)

Important relationships

The elastic string example demonstrates several crucial principles that apply to variable force problems more generally.

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Key relationships for elastic systems:

Potential energy gained equals the total work done in stretching from natural length
The choice of reference point (total length vs extension) doesn't affect the final answer
Integration limits must be adjusted appropriately for your chosen variable definition
The work-energy theorem still applies: work done equals change in kinetic energy plus change in potential energy

Practical tips for exams

Understanding the theory is important, but applying it correctly in exam conditions requires systematic approach and careful attention to detail.

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Exam Success Strategy:

Choose your reference frame carefully - decide whether to use total length or extension as your variable
Set up Hooke's Law correctly for your chosen reference frame
Check your integration limits match your variable definition
Verify your answer makes physical sense - work should be positive for stretching
Show all steps clearly - integration problems require detailed working

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Key Points to Remember:

Work done by a variable force requires integration: $W = \int F(x)dx$
For elastic strings, you can define position as either total length or extension - both give the same result
Hooke's Law must be applied correctly based on your choice of position variable
Potential energy gained equals the total work done in stretching from natural length
Always check that your integration limits match your variable definition
The fundamental principle remains the same: work is force times displacement, but now we must account for changing force

Work Done by a Variable Force (Leaving Cert Applied Maths): Revision Notes