A 67-kg skier grips a moving rope that is powered by an engine and is pulled at constant speed to the top of a 23∘ hill. The skier is pulled a distance x = 300 m along the incline and it takes 2.0 min to reach the top of the hill.
If the coefficient of kinetic friction between the snow and skis is μk = 0.10, what horsepower engine is required if 30 such skiers (max) are on the rope at one time?
Express your answer using two significant figures.

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mickymike92 mickymike92 · Answer 1 · 2022-07-25T20:19:13+02:00

The required horsepower engine is 32 horsepower.

The force of the 30 skiers is calculated as follows:

Mass of the skiers = 30 * 67kg = 2010 kg

Net force acting on the skiers along the x-axis

where f is the frictional force

The kinetic frictional force, f = μN

where

μ = The coefficient of the kinetic friction

N = normal reaction

Net force acting on the skiers along y axis, the

Fy = ma

N = mg cos θ

Substituting for N above

f = μk mg cos θ

Substituting for f in (1)

F = mg sin θ + μk mg cos θ

F = mg(sinθ + μk cos θ)

Work done by the engine in pilling up the skiers, W = Fx

W = mg ( sinθ + μk cos θ)x

x = 300 m

W = (2010 kg) (9.81 m/s²) (sin 23° + (0.10) cos 23°) (300 m)

Work done, W = 2.86 * 10⁶ J

Time taken, t = 2.0 * 60sec = 120 s

1 horsepower = 746 W

Power = 2.86 * 10⁶/ 120 * 1/746

Power = 31.9 horsepower

In conclusion, the power of the engine is the ratio of the work done and time taken.

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