Q5E - University Physics with Modern Physics

Question

A 75.0-kg painter climbs a ladder that is 2.75 m long and leans against a vertical wall. The ladder makes a  $\mathbf{30}\mathbf{°}$ angle with the wall. (a) How much work does gravity do on the painter? (b) Does the answer to part (a) depend on whether the painter climbs at constant speed or accelerates up the ladder?

Step-by-Step Solution

Answer

(a)  $W=-1750J$ 

(b) No

Step 1: Identification of the given data

The given data is listed below as-

• The displacement is, $s=2.75m$
• The mass of the painter is,  $m=75kg$
• The angle that the ladder makes with the wall=  $30°$

Step 2: Significance of the work done on a particle by a constant force

Work done on a particle by a constant force   during a linear displacement   is given by 

 

Now, the force due to gravity on the painter is downwards.

The displacement is along the ladder as shown in fig. below:

Step 3: Determination of work done by gravity on the painter (a)

From the above Fig., it can be seen that

  $\cos 30°=\frac{AC}{AB}$

Here, AC=h and AB=s, then it can be written as below:

  $h=s.\cos 30°$

For  $s=2.75m$

 $$\begin{gathered} h=2.75m\times 0.866 \\ =2.38m \end{gathered}$$

Now, work done by gravity on the painter is given by

  $W=-mgh$

Here, m is the mass of the painter, g is the gravitational constant and h is the height of the wall.

For $m=75kg$ , h=2.38 m and $g=9.8m.s^{-2}$

 $W=-mgh$

  $$\begin{gathered} W=-\left(75kg\right)\times \left(9.8m.s^{-2}\right)\times \left(2.38m\right) \\ =-1750 J \end{gathered}$$

Thus, the total work done by gravity on the painter is -1750 J.

Step 4: To determine whether the painter climbs at constant speed or accelerates up the ladder (b)

Thus, the work done by the gravity on the painter does not depend upon whether the painter climbs at constant speed or accelerates up the ladder.