A ski rope operates on a 15° slope of length 65.0 m. The rope moves at 9.7 km/h and provides power for 40.0 riders at one time, with an average mass per rider of 70 kg. Estimate the power required to operate the tow
To estimate the power required to operate the tow, we can use the formula:
Power = (Force × Distance) ÷ Time
First, we need to calculate the force exerted by the tow rope on the riders. We can do this by considering the forces acting on the riders:
1. Weight force: This is the force due to gravity acting vertically downwards. It can be calculated using the formula: Weight = mass × acceleration due to gravity.
Given that the average mass of each rider is 70 kg and the acceleration due to gravity is approximately 9.8 m/s², we can calculate the weight force.
Weight = 70 kg × 9.8 m/s² = 686 N (Newtons)
2. Component of weight force along the slope: Since the slope is at an angle of 15°, we need to find the component of the weight force acting parallel to the slope. We can calculate this using trigonometry:
Parallel Component = Weight × sin(angle of slope)
Parallel Component = 686 N × sin(15°) = 177.85 N (Newtons)
Next, we calculate the work done by the rope in moving the riders up the slope. The work done is equal to the force applied multiplied by the distance traveled:
Work = Force × Distance
Work = 177.85 N × 65.0 m = 11,549.25 J (Joules)
Now, let's convert the speed of the rope from km/h to m/s:
Speed = 9.7 km/h × (1000 m / 3600 s) = 2.694 m/s
Finally, we can calculate the power required using the formula mentioned earlier:
Power = Work ÷ Time
Power = 11,549.25 J ÷ (65.0 m ÷ 2.694 m/s) = 518.73 W (Watts)
Therefore, the estimated power required to operate the tow is approximately 518.73 Watts.