91Ó°ÊÓ

The height of the ladder is y = 7 m

The mass of the roofer is m = 75 kg

The distance walked by the roofer on the roof of the house is x = 12 m

The work done by an object is equal to the displacement of the object along the direction of the force. This work varies with the direction of the force with displacement.

The work done on the roofer by gravity for climbing up is given as:

$W_{gu}=-mgy\cos \mathrm{θ}$

Here, g is the gravitational acceleration and its value is $9.8m/s^2$, $\mathrm{θ}$ is the angle with the direction of movement of the roofer and its value is$0°$

Substitute all the values in the above equation and we get,

$$\begin{gathered} W_{gu}=-(75 kg)(9.8 m/s^2)(7 m)(\cos 0°) \\ {W}_{gu}=-5145 J \end{gathered}$$

Therefore, the work done on a roofer by gravity is - 5145 J .

The work done on the roofer by gravity for climbing down is given as:

$W_{gd}=mgy\cos \mathrm{θ}$

Here, is the gravitational acceleration and its value is $9.8m/s^2$, $\mathrm{θ}$ is the angle with the direction of movement of the roofer and its value is $0°$

Substitute all the values in the above equation and we get,

$$\begin{gathered} W_{gd}=(75 kg)(9.8 m/s^2)(7 m)(\cos 0°) \\ {W}_{gd}=5145 J \end{gathered}$$

Therefore, the work done on the roofer by gravity for climbing down is 5145 J .

The work done on the roofer by gravity for climbing down is given as:

$W_{rg}=mgx\cos \mathrm{Î\pm }$

Here, g is the gravitational acceleration and its value is $9.8m/s^2$, $\mathrm{Î\pm }$is the angle with the direction of movement of the roofer and its value is $90°$

Substitute all the values in the above equation and we get,

$$\begin{gathered} W_{rg}=(75 kg)(9.8 m/s^2)(12 m)(\cos 90°) \\ {W}_{rg}=0 J \end{gathered}$$

Therefore, the work done on a roofer by gravity for walking on the roof and ground is 0J.

The total work done by gravity on the roofer is given as:

$W=W_{gu}+W_{gd}+W_{rg}$

Substitute all the values in the above equation.

$$\begin{gathered} W=-5145 J+5145 J+0 J \\ W=0 J \end{gathered}$$

Thus, the total work done by gravity on the roofer is 0 J .

The conservative force is the force whose work is independent of the path and only depends on the initial and final position.

The initial and final position of the roofer is the same and the net work done by gravity on the roofer is 0 J . It proves that gravity is a conservative force.