Question

An elevator lifts 6 people to the top of a 100-foot [ft] building in 55 seconds [s].

The empty elevator has a mass of 500 kilograms [kg].
The motor powering the elevator is rated at 8.5 kilowatts (kW) and is 62% efficient.
Determine the average mass of a person who is on the elevator in units of pounds-mass [lbm].

Answer 1

28190 pounds-mass

Step-by-step explanation:

To solve this problem, we need to use several physical principles, such as work, power, energy, and force.

First, we can calculate the work done by the elevator to lift the people and itself to the top of the building. This is given by:

Work = Force x Distance

where the force is equal to the weight of the elevator and the people, and the distance is the height of the building. We can calculate the force as:

Force = (Mass of elevator + Mass of people) x Gravity

where the mass of the elevator is given as 500 kg, the mass of people is unknown, and the gravity is 9.81 m/s^2. We can convert the height of the building to meters as well:

Height = 100 ft x 0.3048 m/ft = 30.48 m

Therefore, the work done by the elevator is:

Work = Force x Distance = (500 kg + 6 x Mass of people) x 9.81 m/s^2 x 30.48 m

Work = (5000 + 58.86 x Mass of people) joules [J]

Next, we can use the power rating of the motor and its efficiency to calculate the energy consumed by the motor during the 55 seconds it takes to lift the elevator. The energy consumed is given by:

Energy = Power x Time / Efficiency

where the power is 8.5 kW, the time is 55 s, and the efficiency is 0.62. We can convert the units of power to joules per second (J/s) as well:

Power = 8.5 kW x 1000 W/kW = 8500 J/s

Therefore, the energy consumed by the motor is:

Energy = Power x Time / Efficiency = 8500 J/s x 55 s / 0.62

Energy = 763709 J

Finally, we can use the conservation of energy to relate the work done by the elevator to the energy consumed by the motor:

Work = Energy

Therefore, we can set the two expressions for work equal to each other and solve for the mass of people:

5000 + 58.86 x Mass of people = 763709 J

Mass of people = (763709 J - 5000) / 58.86 kg = 12778 kg

To convert this mass to pounds-mass, we can use the conversion factor:

1 kg = 2.20462 lbm

Therefore, the average mass of a person on the elevator is:

Mass of people = 12778 kg x 2.20462 lbm/kg = 28190 lbm

So the average mass of a person on the elevator is 28190 pounds-mass.

Answer 2

Approximately 1061.45 pounds-mass (lbm).

Step-by-step explanation:

To determine the average mass of a person on the elevator in pounds-mass (lbm), we need to perform a series of calculations. Let's break it down step by step:

Convert the height of the building from feet (ft) to meters (m):

100 ft = 30.48 m

Convert the time taken to lift the elevator from seconds (s) to hours (h):

55 s = 55/3600 h = 0.0153 h

Calculate the power used by the motor in kilowatts (kW):

Power = Rated Power * Efficiency

Power = 8.5 kW * 0.62 = 5.27 kW

Calculate the work done by the elevator:

Work = Power * Time

Work = 5.27 kW * 0.0153 h = 0.0807 kWh

Convert the work done from kilowatt-hours (kWh) to joules (J):

1 kWh = 3.6 × 10^6 J

Work = 0.0807 kWh * 3.6 × 10^6 J/kWh = 290520 J

Calculate the change in potential energy of the elevator:

Potential Energy = Mass * Gravity * Height

Mass * 9.8 m/s^2 * 30.48 m = 290520 J

Mass = 290520 J / (9.8 m/s^2 * 30.48 m) = 981.22 kg

Subtract the mass of the empty elevator to find the total mass of the people:

Total Mass = Mass - Mass of Empty Elevator

Total Mass = 981.22 kg - 500 kg = 481.22 kg

Convert the total mass of people from kilograms (kg) to pounds-mass (lbm):

1 kg = 2.20462 lbm

Total Mass in lbm = 481.22 kg * 2.20462 lbm/kg ≈ 1061.45 lbm

Therefore, the average mass of a person on the elevator is approximately 1061.45 pounds-mass (lbm).