A force \(\vec{\mathbf{F}} = -K(\hat{x} + \hat{y})\) (where K is a positive constant) acts on a particle moving in the x-y plane. Starting from the origin, the particle is taken along the positive x- axis to the point (a, 0) and then parallel to the y-axis to the point (a, a). The total work done by the forces \vec{F} on the particle is
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For motion of the particle from (0, 0) to (a, 0)
\(\vec{F} = -K(0\hat{i} + a\hat{j}) \Rightarrow \vec{F} = -Ka\hat{j}\)
Displacement \vec{r} = (a\hat{i} + 0\hat{j}) - (0\hat{i} + 0\hat{j}) = a\hat{i}
So work done from (0, 0) to (a, 0) is given by
\(w = \vec{F} \cdot \vec{r} = -Ka\hat{j} \cdot a\hat{i} = 0\)
For motion (a, 0) to (a, a)
\vec{F} = -K(\hat{a}\hat{i} + \hat{a}\hat{j}) and displacement
\vec{r} = (a\hat{i} + a\hat{j}) - (a\hat{i} + 0\hat{j}) = a\hat{j}
So work done from (a, 0) to (a, a) \(W = \vec{F} \cdot \vec{r}\)
\(= -K(\hat{a}\hat{i} + \hat{a}\hat{j}) \cdot \hat{a}\hat{j} = -Ka^{2}\)
So total work done = -Ka^{2}
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