A ski tow operates on a slope of angle 14.9∘ of length 320m . The rope moves at a speed of 12.5km/h and provides power for 55 riders at one time, with an average mass per rider of 68.0kg .Estimate the power required to operate the tow.

o lift the skiers, the rope must do positive work to counteract the negative work developed by the component of the gravitational force acting on the total number of skiers, Frope=Nmgsinα.

SET UP: P=F∥v=Fropev

EXECUTE: Prope=Fropev=[+Nmg(cosϕ)]v.

Prope=[(49riders)(74.0kg)(9.80m/s2)(cos75.2)][(12.6km/h)(1m/s3.60km/h)].

Prope=3.2×104W=32kW.

V = 12.5km/h = 12500m/3600s. = 3.47 m/s.

P = 55F*V = 55*mg*V = 55*68*9.8*3.47 =
127.2 Joules/s = 127.2 Watts.

Correction: Change 127.2 J./s to

127.2 kJ/s = 127.2 KW.

127.2W

133477.338

Well, let's first calculate the time it takes for a rider to be transported up the slope. Given that the rope moves at a speed of 12.5 km/h and the slope's length is 320 m, converting the speed to m/s, we get 12.5 km/h = 12.5 * 1000 m/3600 s ≈ 3.47 m/s.

Now, using this speed, we can calculate the time it takes for a rider to reach the top of the slope by dividing the length of the slope by the speed: 320 m / 3.47 m/s ≈ 92.25 s.

Since the power is the rate at which work is done, we need to find the work done by the ski tow in lifting one rider. The work done is equal to the change in gravitational potential energy, which is given by mgh. But since the slope is at an angle, we need to consider the component of the weight along the slope. Using a bit of trigonometry, we find that the height the rider is lifted is h = 320 m * sin(14.9∘) ≈ 84.48 m.

Now, we can calculate the work done: Work = (mass per rider) * g * h, where g is the acceleration due to gravity.

Plugging in the numbers, we get Work ≈ 68 kg * 9.8 m/s^2 * 84.48 m ≈ 54,451 J.

The time it takes for the ski tow to transfer 55 riders is 55 * 92.25 s ≈ 5,073.75 s.

Finally, the power required to operate the ski tow is given by Power = Work / time.

Plugging in the numbers, we get Power ≈ 54,451 J / 5,073.75 s ≈ 10.73 Watts.

So, the estimated power required to operate the ski tow is approximately 10.73 Watts. But hey, don't underestimate the power of gravity and engineering!

To estimate the power required to operate the ski tow, we need to use the formula:

Power = (Force × Distance) / Time

First, let's calculate the force exerted on the riders. We can use the equation:

Force = Mass × Acceleration

The acceleration can be determined using the slope of the slope, which is given as 14.9 degrees. We can find the component of the gravitational force acting along the slope using trigonometry:

Acceleration = g × sin(θ)

where g is the acceleration due to gravity (approximately 9.8 m/s^2) and θ is the slope angle in radians:

θ = 14.9 degrees = 14.9 × (π/180) radians

Next, we need to calculate the total mass being pulled by the ski tow, which is the product of the number of riders and their average mass:

Total Mass = Number of Riders × Average Mass per Rider

Now, let's calculate the force on the riders:

Force = Total Mass × Acceleration

Next, we need to convert the speed of the moving rope from km/h to m/s:

Speed = 12.5 km/h = 12.5 × (1000/3600) m/s

Finally, we can calculate the required power:

Power = (Force × Distance) / Time

Since we are not given the time, we cannot calculate the exact power required. However, we can assume a standard time, such as one hour, and calculate the power based on that assumption.