A 80 kg skier grips a moving rope that is powered by an engine and is pulled at constant speed to the top of a 25 degree hill. The skier is pulled a distance x = 230m along the incline and it takes 2.3 min to reach the top of the hill. If the coefficient of kinetic friction between the snow and skis is mu_k = 0.15, what horsepower engine is required if 30 such skiers (max) are on the rope at one time?
Since the skier is pulled at a constant speed, the pulling force (for one skier)is given by:
F = m*g[sin theta + mu_k*cos theta]
= 80*10[sin25+0.15*cos25]
= 446.8N (along the incline upwards)
Work done by F in moving distance of 230m:
W=F*x = 446.8*230 =102,776 Joules
Power reqd for this work:
P= W/time = 102,776J/138s = 745 watts = 1HP(approx.)
So for 30 skiers, engine power required = 30HP
Well, let's start by calculating the work done by the engine to pull the skier up the hill. The work done against gravity is given by the formula:
Work_gravity = mgh
where m is the mass of the skier (80 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the hill.
To find h, we can use trigonometry:
h = x * sin(theta)
where x is the distance pulled along the incline (230 m) and theta is the angle of the incline (25 degrees).
Now we can substitute these values into the equation for work:
Work_gravity = (80 kg) * (9.8 m/s^2) * (230 m * sin(25 degrees))
Next, let's calculate the work done against friction. The work done against friction is given by the formula:
Work_friction = mu_k * m * g * d
where mu_k is the coefficient of kinetic friction (0.15), m is the mass of the skier (80 kg), g is the acceleration due to gravity (9.8 m/s^2), and d is the distance pulled along the incline (230 m).
Now we can substitute these values into the equation for work:
Work_friction = (0.15) * (80 kg) * (9.8 m/s^2) * (230 m)
Finally, we can calculate the total work done:
Total_work = Work_gravity + Work_friction
Now, we can calculate the power required by the engine using the formula:
Power = Total_work / time
where time is the time taken to reach the top of the hill (2.3 min = 2.3 * 60 s).
Finally, to find the horsepower, we can convert the power from watts to horsepower:
Horsepower = Power / 746
Now, there you go! With all the calculations done, you have the horsepower required for 30 skiers on the rope at one time. I hope you find this process as thrilling as a skier going down a hill!
To calculate the horsepower required for the engine, we need to consider the work done by the engine to overcome the force of friction and the force due to the incline.
Let's break down the problem step-by-step:
Step 1: Calculate the force due to gravity on the skier
The force due to gravity can be calculated using the formula:
Force due to gravity (Fg) = mass (m) × acceleration due to gravity (g)
Given:
mass of the skier (m) = 80 kg
acceleration due to gravity (g) = 9.8 m/s^2
Calculating:
Fg = 80 kg × 9.8 m/s^2
Fg = 784 N
Step 2: Calculate the force component acting along the incline
The force component acting along the incline, F_Parallel, can be calculated using the formula:
F_Parallel = Fg × sin(theta)
Given:
theta = 25 degrees
Calculating:
F_Parallel = 784 N × sin(25 degrees)
F_Parallel = 336.4 N
Step 3: Calculate the force of kinetic friction
The force of kinetic friction can be calculated using the formula:
F_friction = mu_k × F_N
Given:
coefficient of kinetic friction (mu_k) = 0.15
F_N is the normal force, which is equal to the force due to gravity (Fg) in this case.
Calculating:
F_friction = 0.15 × 784 N
F_friction = 117.6 N
Step 4: Calculate the total force required
The total force required to overcome the force of friction and move the skier up the incline can be calculated as:
F_total = F_Parallel + F_friction
Calculating:
F_total = 336.4 N + 117.6 N
F_total = 454 N
Step 5: Calculate the work done by the engine
The work done by the engine is given by the formula:
Work = force (F_total) × distance (x)
Given:
distance (x) = 230 m
Calculating:
Work = 454 N × 230 m
Work = 104,420 J
Step 6: Convert work to horsepower
Horsepower is a unit used to measure power, defined as 550 foot-pounds per second or 745.7 watts.
To convert work (in joules) to horsepower, we divide the work by the time in seconds (since the power is given in joules per second).
Given:
time (t) = 2.3 min = 2.3 × 60 s
Calculating:
Power = Work / t
Power = 104,420 J / (2.3 × 60 s)
Power ≈ 760 W
To get the horsepower value, we divide the power by 745.7:
Horsepower = Power / 745.7
Horsepower ≈ 760 W / 745.7
Horsepower ≈ 1.02 HP
Step 7: Calculate the horsepower required for 30 skiers
To calculate the total horsepower required if 30 skiers are on the rope at one time, we multiply the horsepower per skier by the number of skiers:
Total Horsepower = Horsepower per skier × Number of skiers
Total Horsepower = 1.02 HP × 30
Total Horsepower = 30.6 HP
Therefore, the horsepower engine required if 30 skiers are on the rope at one time is approximately 30.6 horsepower.
To calculate the horsepower required by the engine, we need to determine the total force acting on the skier as they are pulled up the hill.
First, let's calculate the gravitational force acting on the skier. We can use the formula:
F_gravity = m * g
where
m = mass of the skier = 80 kg
g = acceleration due to gravity = 9.8 m/s^2
F_gravity = 80 kg * 9.8 m/s^2 = 784 N
Next, let's calculate the normal force acting on the skier along the incline. The normal force is equal to the component of the gravitational force perpendicular to the incline. Since the incline is at an angle of 25 degrees, the normal force can be calculated as:
F_normal = F_gravity * cos(theta)
where theta = 25 degrees
F_normal = 784 N * cos(25 degrees) = 704 N
Now, let's calculate the force of friction acting on the skier. The force of friction can be calculated using the equation:
F_friction = mu_k * F_normal
where mu_k = coefficient of kinetic friction = 0.15
F_friction = 0.15 * 704 N = 105.6 N
To overcome the force of friction and move at a constant speed, an equal and opposite force must be exerted on the skier. This force is provided by the tension in the rope, which is generated by the engine.
Next, let's calculate the work done by the engine to move the skier along the incline. The work done is given by the formula:
Work = Force * Distance * cosine(theta)
where
Force = tension in the rope
Distance = 230 m
theta = 25 degrees
Since the skier moves at a constant speed, the work done by the engine is equal to the total work done against all the forces (gravity and friction).
Work = (F_friction + F_gravity) * Distance * cosine(theta)
Work = (105.6 N + 784 N) * 230 m * cos(25 degrees)
Now, let's calculate the time taken to reach the top of the hill. We are given that it takes 2.3 minutes.
Time = 2.3 minutes = 2.3 min * 60 s/min = 138 s
Finally, let's calculate the power (in watts) required by the engine using the formula:
Power = Work / Time
Power = ( (105.6 N + 784 N) * 230 m * cos(25 degrees) ) / 138 s
To convert the power from watts to horsepower, we can use the conversion factor:
1 horsepower = 746 watts
Now, using this formula, you can plug in the values to find the horsepower engine required.