To determine how much work the escalator did on Karen, we need to consider the force exerted by the escalator and the distance she traveled vertically. Work is defined as the product of the force applied and the distance over which that force is applied in the direction of the force. In this case, the escalator is doing work against gravity as it lifts Karen upward.
Understanding the Components of Work
Work can be calculated using the formula:
Work = Force × Distance × cos(θ)
Where:
- Force is the weight of the object (in this case, Karen).
- Distance is the vertical distance the object is lifted.
- θ is the angle between the force and the direction of movement.
Calculating Karen's Weight
First, we need to calculate the force exerted by Karen due to gravity, which is her weight. This can be found using the formula:
Weight = mass × gravity
Given that Karen's mass is 50.1 kg and the acceleration due to gravity is 9.8 m/s², we can calculate her weight:
Weight = 50.1 kg × 9.8 m/s² = 491.98 N
Finding the Vertical Distance
Next, we need to find the vertical distance Karen is lifted by the escalator. Since the escalator is inclined at 29.1 degrees, we can use trigonometry to find the vertical component of the distance traveled:
Vertical Distance = Total Distance × sin(θ)
Substituting the values:
Vertical Distance = 53.8 m × sin(29.1°)
Using a calculator, we find:
Vertical Distance ≈ 53.8 m × 0.485 = 26.1 m
Calculating the Work Done by the Escalator
Now that we have both the force (weight) and the vertical distance, we can calculate the work done by the escalator:
Work = Weight × Vertical Distance
Substituting the values we found:
Work = 491.98 N × 26.1 m ≈ 12885.1 J
Final Result
The escalator did approximately 12885.1 joules of work on Karen as she rode up the longest escalator in the world. This calculation illustrates how forces and distances interact in the context of work, especially in inclined scenarios like escalators.
