Computing Work

Find the work done by the force field \(\mathbf{F}(x,y)=-y\mathbf{i}+x\mathbf{j}\) in moving an object along the half-circle \(C\) traced out by \[ \mathbf{r}(t)=\cos t\,\mathbf{i}+\sin t\,\mathbf{j}\qquad 0\leq t\leq \pi \]

1004

Solution On \(C\), \(x(t)=\cos t\) and \(y(t)=\sin t,\) \(0\leq t\leq \pi\). Then \[ \begin{eqnarray*} \mathbf{F}( x(t),y(t)) & =&-y(t) \mathbf{i}+x(t) \mathbf{j}=-\sin t\,\mathbf{i}+\cos t\,\mathbf{j}\hbox{}\\ d\mathbf{r}(t)& =&( -\sin t\,\mathbf{i}+\cos t\,\mathbf{j}) \,dt \\[4pt] \mathbf{F}\,{\cdot}\, d\mathbf{r}& =&(\sin ^{2}t+\cos ^{2}t)\,dt=dt \end{eqnarray*} \]

The work \(W\) done by \(\mathbf{F}\) is \[ W=\int_{C}\mathbf{F}\,{\cdot}\, d\mathbf{r}=\int_{0}^{\pi }dt=\pi \]