2. A 75.0-kg painter climbs a ladder that is 2.75 m long leaning against a vertical wall. The ladder makes a 30.0° angle with the wall.
(a) How much work does is done by gravity on the painter?
(b) Does the answer to part (a) depend on whether the painter climbs at constant speed or accelerates up the ladder?
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To determine the work done by gravity on the painter, we need to calculate the change in potential energy of the painter as they climb the ladder. The formula for potential energy (PE) is given by PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the change in height.
First, let's find the change in height. Since the ladder is leaning against a vertical wall, the height is the vertical component of the ladder's length. We can find this using trigonometry.
The vertical component (h) can be calculated as h = ladder length * sin(angle).
Given:
Mass (m) = 75.0 kg
Ladder length = 2.75 m
Angle (θ) = 30.0 degrees
Calculating h:
h = 2.75 m * sin(30.0°)
h = 2.75 m * 0.5
h = 1.375 m
Next, we need to substitute the values into the formula for potential energy (PE = mgh).
PE = 75.0 kg * 9.8 m/s^2 * 1.375 m
PE ≈ 1020.75 Joules
Therefore, the work done by gravity on the painter is approximately 1020.75 Joules (a).
Now, let's consider part (b) - whether the answer depends on the painter climbing at a constant speed or accelerating up the ladder.
The work done by gravity on an object only depends on the initial and final positions, not the path taken. In this case, since the painter is climbing vertically up the ladder, the work done by gravity will be the same regardless of whether the painter climbs at a constant speed or accelerates up the ladder.
So the answer to part (a) does not depend on whether the painter climbs at a constant speed or accelerates (b).