Question

Find the work done by force $F=2\hat{i}-3\hat{j}+\hat{k}$ when its point of application moves from the point A(1,2,−3) to the point B(2,0,−5)

Answer 1

Step1: Given data

Force $\overrightarrow{F}=2\hat{i}-3\hat{j}+\hat{k}$ and the moment from point$A(1,2,-2)toB(2,0,-5)$

Step2: Formula used

$W=\overrightarrow{F}·\overrightarrow{s}[W=work,F=force,s=displacement]$

Step3: Calculating work

Converting the points into vectors.

For point $A(1,2,-3)=\hat{i}+2\hat{j}-3\hat{k}$

For point $B(2,0,-5)=2\hat{i}+0\hat{j}-5\hat{k}=2\hat{i}-5\hat{k}$

As we are moving from the point $AtoB$.

Direction vector becomes $(2\hat{i}-5\hat{k})-(\hat{i}+2\hat{j}-3\hat{k})=\hat{i}-2\hat{j}-2\hat{k}$

Hence, $\overrightarrow{s}=(1,-2,-2)$ and $F=2\hat{i}-3\hat{j}+\hat{k}=(2,-3,1)$

Putting the value of force and displacement in the work formula

$$\begin{gathered} W=\overrightarrow{F}·\overrightarrow{s} \\ W=(1i,-2j,-2k)·(2i,-3j,-1k) \\ W=(1\times 2)+(-2\times (-3\left)\right)+\left(\right(-2)\times (-1\left)\right) \\ W=2+6+2 \\ W=6J \end{gathered}$$

Hence, the work done is $6J$.