Electric stairs bring people up from Newyork metro station. Calculate the power consumption of the stairs during the rush hour if the stairs elevate 120 persons per minute up to the street level 30 meters above. Assume that the average weight of passengers is 70 kg and the �efficiency of the stairs is 65%.
To calculate the power consumption of the electric stairs during rush hour, we'll need to determine the work done by the stairs and then convert that to power.
First, let's calculate the work done by the stairs. Work is equal to the force applied multiplied by the distance traveled.
Force = mass * acceleration
Acceleration can be calculated using the following formula:
acceleration = change in velocity / time
Given that the stairs elevate 120 persons per minute, we can calculate the change in velocity using the formula:
change in velocity = distance / time = 30 meters / 60 seconds = 0.5 m/s
Now, let's calculate the force:
mass = number of persons * average weight of a person = 120 * 70 kg = 8400 kg
acceleration = change in velocity / time = 0.5 m/s / 60 s = 0.0083 m/s^2
Force = mass * acceleration = 8400 kg * 0.0083 m/s^2 = 69.72 N
Next, we need to calculate the work done by the stairs:
work = force * distance
work = 69.72 N * 30 m = 2091.6 J (Joules)
Finally, let's calculate the power consumption:
power = work / time
Since we're interested in rush hour, let's assume a rush hour duration of 1 hour (60 minutes):
power = 2091.6 J / 60 min = 34.86 J/min
However, we are given that the efficiency of the stairs is 65%. Efficiency is the ratio of useful work output to the total energy input. In this case, the useful work output is 65% of the total work done by the stairs:
useful work output = efficiency * total work
useful work output = 0.65 * 2091.6 J = 1359.24 J
Therefore, the power consumption of the electric stairs during rush hour, accounting for efficiency, is:
power = useful work output / time
power = 1359.24 J / 60 min = 22.65 J/min