Arrange the four graphs in descending order of total work done; where $$W_1, W_2, W_3$$ and $$W_4$$ are the work done corresponding to figure a, b, c and d respectively.

Here 4 graphs are given and were asked to be arranged in descending order of work done

We know that work done in an F-x graph is equal to the area under the curve.

• Area above x-axis → positive work
• Area below x-axis → negative work
• Net work = (positive area − negative area)

Graph (1);-

Here work done $$W_1$$ is area of this graph which is combination of 2 triangles 

One from $$0\ to\ x_0$$ with a Force of -F
Another from $$x_0\ to\ x_1$$ with a Force of F

Here both are triangles, so area is

 $$Net\ Area\ =\ W_1\ =\ \ \frac{\ 1}{2}\times\ -F\times\ \left(x_0-0\right)\ \ +\ \ \frac{\ 1}{2}\times\ F\times\ \left(x_1-x_0\right)$$

$$Net\ Area\ =\ W_1\ =\ \ -\frac{\ 1}{2}Fx_0\ \ +\ \ \frac{\ 1}{2}\ F\ \left(x_1-x_0\right)$$

Graph (2);-

Here work done $$W_2$$ is area of this graph which is combination of 2 triangles and one rectangle

One triangle from $$0\ to\ x_0$$ with a Force of -F
Another triangle from $$x_0\ to\ x_1$$ with a Force of F
Rectangle from $$x_1to\ x_2$$ with a Force of F

So area is

$$Net\ Area\ =\ W_2\ =\ \ \frac{\ 1}{2}\times\ -F\times\ \left(x_0-0\right)\ \ +\ \ \frac{\ 1}{2}\times\ F\times\ \left(x_1-x_0\right)\ +\ F\times\ \left(x_2-x_1\right)$$

$$Net\ Area\ =\ W_2\ =\ \ -\frac{\ 1}{2}Fx_0\ \ +\ \ \frac{\ 1}{2}\ F\ \left(x_1-x_0\right)\ +\ F\left(x_2-x_1\right)$$

Graph (3);-

Here work done $$W_3$$ is area of this graph which is combination of 3  triangles and one rectangle

First triangle from $$0\ to\ x_0$$ with a Force of -F
Second from $$x_0\ to\ x_1$$ with a Force of F
Third from $$x_2\ to\ x_3$$ with a Force of F
Rectangle from $$x_1to\ x_2$$ with a Force of F


So area is,

$$Net\ Area\ =\ W_3\ =\ \ \frac{\ 1}{2}\times\ -F\times\ \left(x_0-0\right)\ \ +\ \ \frac{\ 1}{2}\times\ F\times\ \left(x_1-x_0\right)\ +\ F\times\ \left(x_2-x_1\right)\ +\ \ \frac{\ 1}{2}\times\ F\times\ \left(x_3-x_2\right)$$

$$Net\ Area\ =\ W_3\ =\ \ -\frac{\ 1}{2}Fx_0\ \ +\ \ \frac{\ 1}{2}\ F\ \left(x_1-x_0\right)\ +\ F\left(x_2-x_1\right)\ +\ \ \frac{\ 1}{2}\ F\ \left(x_3-x_2\right)$$

Graph (4);-

Here work done $$W_4$$ is area of this graph which is combination of 3 triangles and one rectangle

One from $$0\ to\ x_0$$ with a Force of F
Another from $$x_0\ to\ x_1$$ with a Force of -F
Rectangle from $$x_1to\ x_2$$ with a Force of -F
Another from $$x_2\ to\ x_3$$ with a Force of F

So area is,

$$Net\ Area\ =\ W_4\ =\ \ \frac{\ 1}{2}\times\ F\times\ \left(x_0-0\right)\ \ +\ \ \frac{\ 1}{2}\times\ -F\times\ \left(x_1-x_0\right)\ +\ -F\times\ \left(x_2-x_1\right)\ +\ \ \frac{\ 1}{2}\times\ F\times\ \left(x_3-x_2\right)$$

$$Net\ Area\ =\ W_4\ =\ \ \frac{\ 1}{2}\ F\ \left(x_0-0\right)\ \ \ -\ \ \ \frac{\ 1}{2}\ F\ \left(x_1-x_0\right)\ +\ -F\ \left(x_2-x_1\right)\ +\ \ \frac{\ 1}{2}\ F\ \left(x_3-x_2\right)$$