A particle is subject to a force Fx that varies with...

A particle is subject to a force $F_{x}$ that varies with position as in Figure $\mathrm{P} 7.17 .$ Find the work done by the force on the body as it moves (a) from $x=0$ to $x=5.00 \mathrm{~m}$, (b) from $x=5.00 \mathrm{~m}$ to $x=10.0 \mathrm{~m}$, and $(\mathrm{c})$ from $x=$ $10.0 \mathrm{~m}$ to $x=15.0 \mathrm{~m} .$ (d) What is the total work done by the force over the distance $x=0$ to $x=15.0 \mathrm{~m} ?$

Key Concepts

Piecewise Analysis

Piecewise analysis involves breaking down the motion into different segments where the force function may change its properties, such as magnitude or direction. By calculating the work done over each individual segment through integration and then summing these contributions, one obtains the total work performed on the particle over the entire displacement.

Force-Displacement Graph

A force-displacement graph is a graphical representation where the force acting on a particle is plotted as a function of its position. The work done by the force is equivalent to the area under this curve between the specified limits. This visual tool helps in understanding how different ranges of displacement contribute to the total work.

Definite Integration

Definite integration is a mathematical process used to sum an infinite number of infinitesimally small contributions of force over the displacement. It computes the area under the force versus displacement curve, which directly corresponds to the work done by a variable force along a path.

Work

Work is defined as the energy transferred by a force acting over a distance. When a particle moves under the action of a force, the work done by the force is calculated as the product of the force and the displacement over which it is applied. In cases where the force varies with position, the calculation involves integrating the force over the path traveled.

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