An elevator operates on a very slow hydraulic jack. Oil is pumped into a cylinder that contains the piston. The piston lifts the elevator as the internal cylinder pressure (P2) rises in comparison to the outside pressure (P1=101,325 Pa or about 14.7 psi). If the piston can lift 22 kN (about 5000 lb) a vertical distance of 6 m (about 20 ft) in 30 seconds then:
a. Calculate the output power of the piston while it is lifting its maximum load
b. If the efficiency of the piston is 0.9, what is the input power required?
c. What must be the maximum internal pressure (P2) if the total volume of oil required is 0.150 m^3?
a. Output Power = (weight lifted)(height lifted)/time
b. Input power = (Output power)/0.9
c. (P2 - P1)*V = (input power)* time
To answer these questions, we'll need to use principles of fluid mechanics and power calculation. Let's go through each part step by step:
a. To calculate the output power of the piston while it is lifting its maximum load, we need to use the formula:
Output Power = Force x Velocity
We're given the force (22 kN) and the distance traveled (6 m) in the time it takes (30 seconds). First, let's convert the force from kilonewtons to newtons:
Force = 22 kN = 22,000 N
Now let's calculate the velocity of the piston by dividing the distance traveled by the time taken:
Velocity = Distance / Time = 6 m / 30 s = 0.2 m/s
Now we can calculate the output power using the formula:
Output Power = Force x Velocity = 22,000 N x 0.2 m/s = 4,400 W
b. If the efficiency of the piston is 0.9, we need to find the input power required. The efficiency is defined as the ratio of output power to input power:
Efficiency = Output Power / Input Power
Dividing both sides of the equation by the efficiency, we get:
Input Power = Output Power / Efficiency
Input Power = 4,400 W / 0.9 = 4,888.89 W
Therefore, the input power required is approximately 4,888.89 W.
c. To determine the maximum internal pressure (P2) required, we need to use the ideal gas law. However, since the question mentions oil, we'll assume that the oil is incompressible, and its volume remains constant.
Given that the total volume of oil required is 0.150 m^3, and assuming the oil does not compress, we can say that the volume of oil displaced by the piston is equal to the total volume of oil required.
Let's assume the area of the piston is A. Then the volume of oil displaced by the piston is given by:
Volume = A x Distance = A x 6 m
We're also given the force exerted by the piston, which is the product of the pressure and the area:
Force = Pressure x Area
Rearranging the equation, we can find the pressure:
Pressure = Force / Area
Now we can substitute the volume equation into the pressure equation:
Pressure = (Force / Area) = (Force / Area) x (Distance / Distance) = (Force x Distance) / (Area x Distance)
Since Volume = A x Distance, we can rewrite it as:
Pressure = (Force x Distance) / Volume
Now let's substitute the given values:
Pressure = (22,000 N x 6 m) / 0.150 m^3 = 880,000 N/m^3
Therefore, the maximum internal pressure required (P2) is approximately 880,000 N/m^3.