Question

The force acting on a particle varies as shown in Figure P7.14. Find the work done by the force on the particle as it moves

(a) from X=0 to X=8m,

(b) from X=8m to X10m, and

(c) from X=0 to X=10m.

Short answer

(a) Work done between X=0 to X=8m is 24J.

(b) Work done betweenX=8m to X10m is -2J.

(c) Work done between X=0 to X=10m is 22J.

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=0 to X=8m

Label the figure as follows

From the labeled figure, the work done from X=0 to X=8 is the area of the triangle ABD calculated as follows

$\begin{array}{c}W=\frac{1}{2}\times BF\times AC\\ =\frac{1}{2}\times 6\mathrm{N}\times 8\mathrm{m}\\ =24·\left(1\mathrm{N}·\mathrm{m}\times \frac{1\mathrm{J}}{1\mathrm{N}·\mathrm{m}}\right)\\ =24\mathrm{J}\end{array}$

The required work done is 24J.

Determining the work done from X=8m to X=10m

From the labeled figure, the work done from X=8m to X=10m is the area of the triangle CDE calculated as follows

$\begin{array}{c}W=\frac{1}{2}\times GD\times CE\\ =\frac{1}{2}\times \left(-2\mathrm{N}\right)\times \left(10-8\right)\mathrm{m}\\ =-2·\left(1\mathrm{N}·\mathrm{m}\times \frac{1\mathrm{J}}{1\mathrm{N}·\mathrm{m}}\right)\\ =-2\mathrm{J}\end{array}$

The required work done is -2J.

Determining the work done from X=0 to X=10m

Work done from X=0 to X=10m is the sum of work done from X=0 to X=8m and the work done from X=8m to X=10m. Hence the total work done is

$\begin{array}{c}W=24\mathrm{J}-2\mathrm{J}\\ =22\mathrm{J}\end{array}$

The required work done is 22J.