Question

An object is displaced from position vector vec(r)_(1) = (2 hat(i) + 3 hat(j)) m to vec(r)_(2) == (4 hat(j) + 6 hat(k))m under a force vec(F) = (3 x^(2) hat(i) + 2y hat(j)) N. Find the. work done this force.

Answer 1

Given :-

$\vec{r}_{1}$= (2î + 3j) m

$\vec{r}_{2}$= (4î + 6j) m

F = (3x² î + 2y j) N.

As we know that,

W = $\int^{{r}_{2}}_{{r}_{1}}{F•dr}$

W = $\int^{{r}_{2}}_{{r}_{1}}{(3{x}^{2}dx+2y\:dy)}$

W = $[{x}^{3}+{y}^{2}]^{{r}_{2}}_{{r}_{1}}$

W = [(4)³-(2)³] + [(6)²-(3)²]

W = [64-8]+[36-9]

W = 83 J.

In the above, we saw that while calculating the work done we didn't mention the path through which the object was displaced.

Only Initial and Final coordinates were required.

It shows that, the work done is path independent or work done will be equal to whichever path we follow.