Question

A particle is subject to a force ${\mathit{F}}_{\mathbf{x}}$that varies with position as shown in Figure P7.15. Find the work done by the force on the particle as it moves

(a) from X=0m to X=5m,

(b) from X=5m to X=10m, and

(c) from 10m to X=15m.

(d) What is the total work done by the force over the distance X=0m to X=15m?

Short answer

(a) Work done between X=0m and X=5m is 7.5J.

(b) Work done between X=5mand X10m is 15J.

(c) Work done between X=10m and X15mis 7.5J.

(c) Work done between X=0m and X=15m is 30J.

Step-by-step solution

Given data

A force-displacement graph is provided.

Work done from force-displacement graph

Work done is the area under the force-displacement graph.

Determining the work done from X=0m to X=5m

Label the figure as follows

From the labeled figure, the work done from X=0m to X=5m is the area of the triangle ABF calculated as follows

$\begin{array}{c}W=\frac{1}{2}\times BF\times AF\\ =\frac{1}{2}\times 3\mathrm{N}\times 5\mathrm{m}\\ =7.5·\left(1\mathrm{N}·\mathrm{m}\times \frac{1\mathrm{J}}{1\mathrm{N}·\mathrm{m}}\right)\\ =7.5\mathrm{J}\end{array}$

The required work done is 7.5J.

Determining the work done from X=5m to X=10m

From the labeled figure, the work done from X=5m to X=10m is the area of the rectangle BCEF calculated as follows

$\begin{array}{c}W=BF\times EF\\ =\left(3\mathrm{N}\right)\times \left(10-5\right)\mathrm{m}\\ =15·\left(1\mathrm{N}·\mathrm{m}\times \frac{1\mathrm{J}}{1\mathrm{N}·\mathrm{m}}\right)\\ =15\mathrm{J}\end{array}$

The required work done is 15J.

Determining the work done from X=10m to X=15m

From the labeled figure, the work done from X=10m to X=15m is the area of the triangle CDE calculated as follows

$\begin{array}{c}W=\frac{1}{2}\times CE\times DE\\ =\frac{1}{2}\times 3\mathrm{N}\times 5\mathrm{m}\\ =7.5·\left(1\mathrm{N}·\mathrm{m}\times \frac{1\mathrm{J}}{1\mathrm{N}·\mathrm{m}}\right)\\ =7.5\mathrm{J}\end{array}$

The required work done is 7.5J.

Determining the work done from X=0m to X=15m

Work done from X=0m to X=15m is the sum of the areas of triangle ABF, rectangle BCEF and triangle CDE calculated as follows

$\begin{array}{c}W=7.5\mathrm{J}+15\mathrm{J}+7.5\mathrm{J}\\ =30\mathrm{J}\end{array}$

The required work done is 30J.