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When dealing with forces that vary with displacement, we cannot use the equation Work = Force × Displacement to calculate the work done by the general variable force. This is because this equation assumes that the force is constant. Instead, we must consider how the force changes as an object moves.
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One way of calculating the work done is to divide the displacement into several very small steps Δx, find force F at each step then calculate the work done at each step and finally add up the work done at each step to find the total work done.

The Physics of Work
Work, as defined in physics, quantifies energy transfer when an object moves due to an external force. It's a fundamental aspect of energy and work in physics, typically expressed by the equation W = F * d, where W represents work, F is the force applied, and d is the displacement.

Variable and Constant Forces
While the formula W = F * d is straightforward for constant forces, it requires modification when dealing with variable forces. This scenario is common in real-world applications, such as when considering the work done by spring force, a classic example of a force that changes with displacement.

General Variable Force
When forces vary as an object moves, like in the case of a spring (Hooke's law), the simple product of force and displacement no longer suffices. Instead, we examine small displacements, where the force can be approximated as constant, and sum the work over these increments.

Calculating Work for Variable Forces
The work done by a variable force is equivalent to the area under the force-displacement graph. For infinitesimally small segments of displacement, this area—and thus the work done—can be determined using integral calculus, where the work is the integral of force over the displacement from the initial position x1 to the final position x2.

Practical Application: Work Done by a Spring Force
To illustrate, let's consider a spring force, where the force varies as a function of displacement (F = -kx, with k being the spring constant). The work done by such a force between two points can be calculated using the integral of the force function, providing a concrete example of the work-energy principle in action.

From Theory to Practice: A Solved Example
Consider a force that changes with the square of displacement (F = 3x^2). To find the work done from x = 1 m to x = 3 m, we integrate the force function across this range, yielding the work W = ∫(3x^2) dx from x = 1 to x = 3, which simplifies to W = x^3 from 1 to 3, or W = 27 - 1 = 26 J.

SI Units and Their Importance
In physics, adhering to the SI unit system is critical. For work, the SI unit is the joule (J), which is pivotal when discussing work done by a force. This consistent use of units ensures clarity and accuracy in calculations and communication within the scientific community.

Summary: Work Done by a General Variable Force
Calculating the work done by a variable force across a displacement involves integrating the force function with respect to displacement. For a variable force described by F(x), where F changes with x, the work done from an initial point x_initial to a final point x_final is given by W = ∫ from x_initial to x_final of F(x) dx. This method quantifies work as the area under the force-displacement curve.

Key Moments:
0:00 Understanding Work Done by Variable Forces: Introducing the concept of work in physics when dealing with forces that vary with displacement.
1:04 Calculating Work with Variable Forces: Exploring the methodology for calculating work when forces are not constant, using the concept of infinitesimally small displacements.
2:08 Segment-Wise Work Calculation: Demonstrating how work is calculated for each small segment of displacement and how these calculations contribute to understanding the total work done.
3:21 Integrating Work for Variable Forces: Introducing the integral method to sum up the work done over all segments for a precise total, highlighting the transition from a summation approach to integration.
5:02 Real-World Application: Applying the theoretical concepts to calculate the work done by a variable force in a practical example, enhancing understanding through direct application.

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