Physics

Physics

Problem

Which of the following mathematical expressions represents the work done by a force that varies with position x when the particle moves from xix_i to xfx_f along the x-axis? a) W=∫xixfx dF\displaystyle W = \int_{x_i}^{x_f} x\,dF b) W=F(xi)+F(xf)2 (xf−xi)\displaystyle W = \dfrac{F(x_i)+F(x_f)}{2}\,(x_f-x_i) c) W=∫xixfF(x) dx\displaystyle W = \int_{x_i}^{x_f} F(x)\,dx d) W=F(xf) (xf−xi)\displaystyle W = F(x_f)\,(x_f-x_i)
Intuitive explanation

Work done by a variable force is fundamentally about summing up the tiny contributions F(x) dxF(x) \, dx as the force changes along the path. Instead of multiplying a constant force by distance, you need to integrate the force function over the interval—this captures how the force's strength changes at each point. The integral ∫xixfF(x) dx\int_{x_i}^{x_f} F(x) \, dx is your tool for turning a continuously changing force into a total work value.

Intuitive explanation

Work done by a variable force is fundamentally about summing up the tiny contributions F(x) dxF(x) \, dx as the force changes along the path. Instead of multiplying a constant force by distance, you need to integrate the force function over the interval—this captures how the force's strength changes at each point. The integral ∫xixfF(x) dx\int_{x_i}^{x_f} F(x) \, dx is your tool for turning a continuously changing force into a total work value.

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