Question 13.83: (a) Calculate the work done from D to O by the force P of Pr...

From solution to (a) of Problem 13.82

\mathbf{P}=\frac{x \mathbf{i}+y \mathbf{j}+z \mathbf{k}}{\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2}}

(a) U_{O D}=\int_{O}^{D} \mathbf{P} \cdot d \mathbf{r}

\begin{aligned}\mathbf{r} & =x \mathbf{i}+y \mathbf{j}+z \mathbf{k} \\d \mathbf{r} & =d x \mathbf{i}+d y \mathbf{j}+d z \mathbf{k} \\\mathbf{P} & =\frac{x \mathbf{i}+y \mathbf{j}+z \mathbf{k}}{\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2}}\end{aligned}

Along the diagonal. \quad x=y=z

Thus, \quad\mathbf{P} \cdot d r =\frac{3 x}{\left(3 x^{2}\right)^{1 / 2}}=\sqrt{3}

U_{O-D}  =\int_{0}^{a} \sqrt{3} d x=\sqrt{3} a\quad\quad U_{O D}=\sqrt{3} a \blacktriangleleft

(b) \quad U_{O A B D O}  =U_{O A B D}+U_{D O}

From Problem 13.82

U_{O A B D}=\sqrt{3}a  at left 

The work done from D to O along the diagonal is the negative of the work done from O to D.

U_{D O}=-U_{O D}=-\sqrt{3} a \quad[\text { see part }(a)]

Thus,

U_{O A B D O}=\sqrt{3} a-\sqrt{3} a=0\blacktriangleleft