A skier of mass 72 kg is pulled up a slope by a motor-driven cable.
(a) How much work is required to pull him 65 m up a 30° slope (assumed frictionless) at a constant speed of 2.7 m/s?
(b) What power must a motor have to perform this task?
(a) Work = M g H, where the height of the rise (H) is 65 sin 30 = 32.5 m
(b) Power = (Work)/(Time)
Time = 65m/2.7 m/s = 24.07 s
To solve this problem, we will need to use the concepts of work and power.
(a) Calculate the work required to pull the skier up the slope:
The work done is given by the formula: work = force × distance × cosine(theta)
In this case, the force required to pull the skier up the slope is equal to the component of the gravitational force acting along the slope, which can be calculated as: force = mass × acceleration
The acceleration can be calculated using the formula: acceleration = gravity × sine(theta), where theta is the angle of inclination.
Let's calculate:
Mass (m) = 72 kg
Distance (d) = 65 m
Inclination (theta) = 30°
Speed (v) = 2.7 m/s
Gravitational acceleration (g) = 9.8 m/s²
Acceleration (a) = g × sine(theta)
= 9.8 m/s² × sine(30°)
Force (F) = m × a
= 72 kg × (9.8 m/s² × sine(30°))
Now we need to find the work done by the force, which is given by:
Work (W) = F × d × cosine(theta)
= 72 kg × (9.8 m/s² × sine(30°)) × 65 m × cosine(30°)
Calculate this value to get the work required.
(b) Calculate the power required for the motor:
Power is defined as the rate at which work is done. It can be calculated using the formula:
Power (P) = work / time
In this case, since the skier is being pulled up at a constant speed, the time taken is given by distance / speed.
Power (P) = work / (distance / speed)
Substitute the values of work and distance into the formula to calculate the power required.
Let's calculate:
Power (P) = Work / (Distance / Speed)
Now you can calculate the power required by the motor.
To calculate the work required to pull the skier up the slope, we can use the formula:
Work = Force * Distance * cosine(theta)
Where:
- Force is the gravitational force acting on the skier, which can be calculated as the product of mass and acceleration due to gravity.
- Distance is the displacement along the slope.
- Theta is the angle of the slope.
Let's start by calculating the force applied to the skier using the formula:
Force = mass * acceleration due to gravity
Given:
- Mass (m) = 72 kg
- Acceleration due to gravity (g) = 9.8 m/s^2
Force = 72 kg * 9.8 m/s^2 = 705.6 N
Now, we can calculate the work using the formula mentioned earlier:
Work = 705.6 N * 65 m * cosine(30°)
To find the cosine of 30°, we need to use a scientific calculator or lookup table, which gives us:
cos(30°) ≈ 0.866
Work = 705.6 N * 65 m * 0.866 = 31,136.64 Joules
So, the work required to pull the skier up the slope is approximately 31,136.64 Joules.
To calculate the power required by the motor, we can use the formula:
Power = Work / Time
Since the skier is being pulled up the slope at a constant speed of 2.7 m/s, we can calculate the time required using the formula:
Time = Distance / Speed
Time = 65 m / 2.7 m/s ≈ 24.07 seconds
Now, we can calculate the power using the formula mentioned earlier:
Power = 31,136.64 Joules / 24.07 seconds
Power ≈ 1,293.68 Watts
Therefore, the motor should have a power output of approximately 1,293.68 Watts to perform this task.