A T-bar tow is planned in a new ski area. At any one time, it will be required, to pull a maximum of 80 skiers up a 600-m slope inclined at 15 degrees above the horizontal at a speed of 2.5 m/s. The coefficient of kinetic friction between the skiers skis and the snow is typically 0.960. As the manager of the facility, what motor power should you request of the construction contractor if the mass of the average skier is 75 kg. Assume you want to be ready for any emergency and will order a motor whose power rating is 50 percent larger than the bare minimum.
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To find the motor power required for the T-bar tow, we need to calculate the total force needed to pull the skiers up the inclined slope at the given speed.
Step 1: Determine the gravitational force acting on the skiers.
The gravitational force (F_gravity) can be calculated using the formula:
F_gravity = mass * acceleration due to gravity
Given that the average mass of each skier is 75 kg and the acceleration due to gravity is approximately 9.8 m/s², we can calculate the force:
F_gravity = 75 kg * 9.8 m/s²
Step 2: Determine the frictional force opposing the motion.
The coefficient of kinetic friction (μ) between the skiers' skis and the snow is given as 0.960. The frictional force (F_friction) can be calculated using the formula:
F_friction = coefficient of kinetic friction * F_normal
The normal force (F_normal) can be calculated using the formula:
F_normal = mass * gravitational acceleration * cos(theta)
Given that the slope is inclined at 15 degrees above the horizontal, we can calculate the normal force and then the frictional force:
F_normal = 75 kg * 9.8 m/s² * cos(15 degrees)
F_friction = 0.960 * F_normal
Step 3: Determine the net force required to pull the skiers uphill.
The net force required is the sum of the gravitational force and the frictional force:
F_net = F_gravity + F_friction
Step 4: Determine the power required to achieve the desired speed.
The power (P) can be calculated using the formula:
P = F_net * velocity
Given that the slope has a length of 600 m and the desired speed is 2.5 m/s, we can calculate the power required:
P = F_net * 2.5 m/s
Step 5: Account for emergency situations and order a motor with a 50% power increment.
To ensure you are prepared for any emergency situations, you should order a motor with a power rating 50% larger than the bare minimum. This means multiplying the calculated power by 1.5:
P_final = P * 1.5
Now you have the motor power (P_final) that you should request from the construction contractor to meet the requirements of the T-bar tow in the new ski area.