To calculate the work done by gravity on the painter, we can use the following formula for work:

$W = F \cdot d \cdot \cos(\theta)$

Where:

• $W$ is the work done by gravity.
• $F$ is the force due to gravity, which is the weight of the painter.
• $d$ is the vertical displacement of the painter.
• $\theta$ is the angle between the force and the displacement direction.

Step 1: Force due to gravity

The weight of the painter is given by: $F = m \cdot g$ Where:

• $m = 74.0 \, \text{kg}$ is the mass of the painter.
• $g = 9.81 \, \text{m/s}^2$ is the acceleration due to gravity.

$F = 74.0 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 725.94 \, \text{N}$

Step 2: Calculate the vertical displacement $d$

Since the ladder makes a $25^\circ$ angle with the wall, we can find the vertical displacement using trigonometry. The vertical displacement is the height of the painter, which corresponds to the adjacent side of the triangle formed by the ladder and the wall:

$d = L \cdot \cos(\theta)$ Where:

• $L = 2.25 \, \text{m}$ is the length of the ladder.
• $\theta = 25^\circ$ is the angle with the wall.

$d = 2.25 \, \text{m} \cdot \cos(25^\circ) = 2.25 \, \text{m} \cdot 0.9063 = 2.04 \, \text{m}$

Step 3: Work done by gravity

Gravity acts vertically downwards, so the angle between the force of gravity and the vertical displacement is $0^\circ$ (since they are in the same direction). Hence, $\cos(0^\circ) = 1$.

The work done by gravity is: $W = F \cdot d \cdot \cos(0^\circ) = 725.94 \, \text{N} \cdot 2.04 \, \text{m} \cdot 1 = 1480.92 \, \text{J}$

Thus, the work done by gravity on the painter is approximately: $W \approx 1480.9 \, \text{J}$

Let me know if you'd like further explanation or additional questions!

5 Related Questions:

1. How would the work done by gravity change if the ladder was steeper (a larger angle)?
2. What would happen to the work if the painter was heavier?
3. How would you calculate the work done if the painter descended the ladder instead?
4. What role does the angle between the ladder and wall play in determining the vertical displacement?
5. How does gravitational potential energy relate to the work done by gravity?

Tip:

Work is always a scalar quantity, and only the component of force in the direction of displacement contributes to the work done.