Evaluate the work done between point 1 and point 2 for the conservative field F.
-$$\mathrm { F } = 5 x \mathbf { i } + 5 y \mathbf { j } + 5 \mathrm { z } \mathbf { k } ; \mathrm { P } _ { 1 } ( 2,2,3 ) , \mathrm { P } _ { 2 } ( 6,7,9 )$$
A) $\mathrm { W } = \frac { 915 } { 2 }$
B) $W = - \frac { 745 } { 2 }$
C) $W = 0$
D) $\mathrm { W } = \frac { 745 } { 2 }$

Question

Quizplus · Accepted Answer

The work done between point 1 and point 2 for a conservative field is independent of the path taken and depends only on the initial and final points. Therefore, we can use the potential function of the field to calculate the work. The potential function, ϕ, can be found by integrating the components of F with respect to their respective variables.

$$\phi( x , y , z ) = \int { 5x dx } + \int { 5y dy } + \int { 5z dz } = \frac{5x^2}{2} + \frac{5y^2}{2} + \frac{5z^2}{2} + C  $$

where C is the constant of integration.

Next, we can calculate the work done by finding the potential difference between the two points and using the formula:

$$ W = \phi (P_2) - \phi(P_1) $$

\begin{align*}
W &= \left( \frac{5(6)^2}{2} + \frac{5(7)^2}{2} + \frac{5(9)^2}{2} + C \right) - \left( \frac{5(2)^2}{2} + \frac{5(2)^2}{2} + \frac{5(3)^2}{2} + C \right)\
&= \frac{745}{2}
\end{align*}

Therefore, the correct answer is D) W = 745/2.

Evaluate the Work Done Between Point 1 and Point 2