Question

Calculate the work done if a particle is moved along a path AB−BC−CD−DE−EF−FA, as shown in the figure in presence of a force $F=\left(\alpha y\hat{i}+2\alpha x\hat{j}\right)N$ , where x and y are in meters and $\alpha =-1N{m}^{-1}$.

Answer 1

Step 1. Given data

Force, $F=\left(\alpha y\hat{i}+2\alpha x\hat{j}\right)N$

$\alpha =-1N{m}^{-1}$

Step 2. Calculating the work done from A to B, $W_{A\to B}$

By using the formula of work, $W$

$W=\int F.dr$

$W_{A\to B}=\int \alpha y.dx$

$W_{A\to B}=\alpha \times 1{\int }_0^{1}dx$

$W_{A\to B}=-1$ [$eq\left(1\right)$]

Step 2. Calculating the work done from B to C, $W_{B\to C}$

$W_{B\to C}=2\alpha \times 1{\int }_1^{0.5}dy$

$W_{B\to C}=1$ [$eq\left(2\right)$]

Step 3. Calculating the work done from C to D, $W_{C\to D}$

$W_{C\to D}=\alpha \times 1{\int }_1^{0.5}dx$

$W_{C\to D}=\frac{1}{4}$ [$eq\left(3\right)$]

Step 3. Calculating the work done from D to E, $W_{D\to E}$

$W_{D\to E}=2\alpha {\int }_{0.5}^{0}xdy$

$W_{D\to E}=\frac{1}{2}$ [$eq\left(4\right)$]

Step 4. Calculating the work done from E to F and F to A

Work done from $E$ to $F$ and $F$ to $A$ is zero as $x=0andy=0$.

Step 4. Calculating the total work done, $W$

Adding all the equations, we get

$W=0.75J$

Therefore, the work done, $W=0.75J$