A skier of mass 67.0 kg is pulled up a slope by a motor-driven cable. How much work is required to pull the skier 59 m up a 32° slope (assumed to be frictionless) at a constant speed of 2.0 m/s?
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To calculate the work required to pull the skier up the slope, we need to use the equation:
Work = Force × Distance × cosine(theta)
Where:
- Force is the force applied to pull the skier uphill.
- Distance is the distance the skier travels uphill.
- theta is the angle between the force applied and the direction of motion (slope's angle).
In this case, we need to break down the force into its vertical and horizontal components. The vertical component is the weight of the skier, which can be calculated using the equation:
Force_vertical = mass × gravity
Where:
- mass is the mass of the skier.
- gravity is the acceleration due to gravity, approximately 9.8 m/s^2.
The horizontal component of the force is responsible for the skier's motion along the slope. Therefore, we calculate it using the equation:
Force_horizontal = Force_vertical × sin(theta)
Finally, we can calculate the work using the formula mentioned above:
Work = Force_horizontal × Distance × cosine(theta)
Given:
- mass = 67.0 kg
- Distance = 59 m
- theta = 32°
- speed = 2.0 m/s
- gravity = 9.8 m/s^2
Let's calculate the work:
Step 1: Calculate the force_vertical
Force_vertical = mass × gravity
Force_vertical = 67.0 kg × 9.8 m/s^2
Force_vertical = 657.6 N
Step 2: Calculate the force_horizontal
Force_horizontal = Force_vertical × sin(theta)
Force_horizontal = 657.6 N × sin(32°)
Force_horizontal = 657.6 N × 0.5299
Force_horizontal = 348.4 N
Step 3: Calculate the work
Work = Force_horizontal × Distance × cosine(theta)
Work = 348.4 N × 59 m × cos(32°)
Work = 348.4 N × 59 m × 0.8480
Work = 17400 J (Joules)
Therefore, the work required to pull the skier up the slope is 17400 Joules.