Question

Find the work done by a force \({\bf{F}} = 8{\bf{i}} - 6{\bf{j}} + 9{\bf{k}}\) that moves an object from the point \((0,10,8)\) to the point \((6,12,20)\) along a straight line. The distance is measured in meters and the force in newtons.

Short answer

Work done is\(144\;{\rm{J}}\).

Step-by-step solution

Formula used to find the work done by given force that moves an object from onepoint to the other along a straight line

The equation to find the work done\((W)\)is,

\(W = {\rm{F}} \cdot {\rm{D}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.........\left( 1 \right)\)

Here,

\({\rm{F}}\) is the Force, and

\({\rm{D}}\)is the displacement vector that is\(\overrightarrow {PQ} \).

Find the work done by given force

Given: \({\rm{F}} = 8{\rm{i}} - 6{\rm{j}} + 9{\rm{k}}\)

The object moves along the straight line from \((0,10,8)\)to \((6,12,20)\).

Consider the points \(P(0,10,8)\)and\(Q(6,12,20)\).

Find \(\overrightarrow {PQ} \)

\(\begin{aligned}{c}\overrightarrow {PQ} &= Q(6,12,20) - P(0,10,8)\\ &= (6 - 0){\rm{i}} + (12 - 10){\rm{j}} + (20 - 8){\rm{k}}\\ &= 6{\rm{i}} + 2{\rm{j}} + 12{\rm{k}}\end{aligned}\)

In equation (1), substitute \(6{\rm{i}} + 2{\rm{j}} + 12{\rm{k}}\) for \({\rm{D}}\) and \(8{\rm{i}} - 6{\rm{j}} + 9{\rm{k}}\)for \({\rm{F}}\).

\(\begin{aligned}{c}W &= (8{\rm{i}} - 6{\rm{j}} + 9{\rm{k}}) \cdot (6{\rm{i}} + 2{\rm{j}} + 12{\rm{k}})\\ &= 48 - 12 + 108\\ &= 144\;{\rm{J}}\end{aligned}\)

Thus, the work done is\(144\;{\rm{J}}\).