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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 9: Q. 42 (page 613)

In the definition of work given in this section, what is the work done if F is orthogonal to $\overset{鈫�}{A B} ?$

Short Answer

Expert verified

There will be no work done ifFis orthogonal to $\overset{鈫�}{A B} .$

Step by step solution

01

Step 1. Given Information

We have to find what will be the work done ifFis orthogonal to $\overset{鈫�}{A B} .$

02

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 $W = F 路 \overset{鈫�}{A B} .$

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 $胃 .$

We know the work done is localid="1646819766257" $W = F 路 \overset{鈫�}{A B} . . . . . . (a)$

As we know if uand vare two nonzero vectors, the angle $胃, 0 鈮� 胃鈮� 蟺,$ between uand vis determined by the formula

$\cos 胃 = \frac{u 路 v}{||u|| ||v||}$

Thus, $\cos 胃 = \frac{F 路 \overset{鈫�}{A B}}{||F|| ||\overset{鈫�}{A B}||}$

03

Step 3. F is orthogonal to AB→

So, $胃 = 90^{鈭�}$

$\cos 90^{鈭�} = \frac{F 路 \overset{鈫�}{A B}}{||F|| ||\overset{鈫�}{A B}||} 0 = F 路 \overset{鈫�}{A B} . . . . . . . . (b)$

From (a) and (b) we get,

$W = 0$

Therefore there will be no work done if F is orthogonal to $\overset{鈫�}{A B} .$

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