Question

In the definition of work given in this section, what is the work done if F is orthogonal to$\overrightarrow{AB}?$

Short answer

There will be no work done ifFis orthogonal to$\overrightarrow{AB}.$

Step-by-step solution

Step 1. Given Information

We have to find what will be the work done ifFis orthogonal to$\overrightarrow{AB}.$

Step 2. Finding the work done

As we know the work done by a constant force Fin moving an object from a point, A to a point B is defined as $W=\mathbf{F}·\overrightarrow{AB}.$

Now, let F constant force is moving an object from point Ato point B,and the angle between the force vector and displacement vector is $\theta .$

We know the work done is localid="1646819766257" $W=\mathbf{F}·\overrightarrow{AB}......\left(a\right)$

As we know if uand vare two nonzero vectors, the angle $\theta ,0\le \theta \le \mathrm{\pi },$ between uand vis determined by the formula

$\cos \theta =\frac{\mathit{u}\mathbf{·}\mathit{v}}{\left|\left|u\right|\right|\left|\left|v\right|\right|}$

Thus,$\cos \theta =\frac{\mathbf{F}·\overrightarrow{AB}}{\left|\left|\mathbf{F}\right|\right|\left|\left|\overrightarrow{AB}\right|\right|}$

Step 3. F is orthogonal to AB→

So, $\theta ={90}^{\circ }$

$$\begin{gathered} \cos {90}^{\circ }=\frac{\mathbf{F}·\overrightarrow{AB}}{\left|\left|\mathbf{F}\right|\right|\left|\left|\overrightarrow{AB}\right|\right|} \\ 0=\mathbf{F}·\overrightarrow{AB}........\left(b\right) \end{gathered}$$

From (a) and (b) we get,

$W=0$

Therefore there will be no work done if F is orthogonal to$\overrightarrow{AB}.$