Problem 67 - College Physics

Question

\(\bullet\) \(\bullet\) An elevator has mass \(600 \mathrm{kg},\) not including passengers. The elevator is designed to ascend, at constant speed, a vertical distance of 20.0 \(\mathrm{m}\) (five floors) in \(16.0 \mathrm{s},\) and it is driven by a motor that can provide up to 40 hp to the elevator. What is the maximum number of passengers that can ride in the elevator? Assume that an average passenger has mass 65.0 \(\mathrm{kg}\) .

Step-by-Step Solution

Answer

The maximum number of passengers is 28.

Step 1: Calculate the Power Used

First, we calculate the power used to lift the elevator without any passengers. The power is given by the equation:\[P = \frac{W}{t}\]where the work done \(W\) is the force times distance (\(F \times d\)) and force \(F\) is \(m \cdot g\) (mass times gravitational acceleration). So we have:\[W = m \cdot g \cdot d = 600 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 20 \, \text{m} = 117600 \, \text{J} \]Divide by time:\[P_{\text{used}} = \frac{117600 \, \text{J}}{16 \, \text{s}} = 7350 \, \text{W} \]

Step 2: Convert Motor Power to Watts

The motor's power capability is given in horsepower (hp). We need to convert 40 hp to watts using the conversion: 1 hp = 746 W.\[P_{\text{motor}} = 40 \, \text{hp} \times 746 \, \text{W/hp} = 29840 \, \text{W} \]

Step 3: Calculate Extra Power Available

The extra power available to lift passengers is the motor power minus the power used for the empty elevator:\[P_{\text{extra}} = P_{\text{motor}} - P_{\text{used}} = 29840 \, \text{W} - 7350 \, \text{W} = 22490 \, \text{W} \]

Step 4: Determine Additional Mass that Can Be Lifted

This extra power can be used to lift additional mass of passengers. The power needed to lift is given by:\[P = \frac{m_{\text{pass}} \cdot g \cdot d}{t} = m_{\text{pass}} \cdot 9.8 \, \text{m/s}^2 \cdot \frac{20 \, \text{m}}{16 \, \text{s}} = 12.25 \, m_{\text{pass}} \, \text{W} \]We solve for \(m_{\text{pass}}\):\[m_{\text{pass}} = \frac{P_{\text{extra}}}{12.25} = \frac{22490 \, \text{W}}{12.25 \, \text{W/kg}} = 1835.92 \, \text{kg} \]

Step 5: Calculate Maximum Number of Passengers

Since each passenger has a mass of 65 kg, we can calculate the number of passengers by dividing the total extra mass that can be lifted by the mass of one passenger:\[N_{\text{pass}} = \frac{m_{\text{pass}}}{65 \, \text{kg/passenger}} = \frac{1835.92 \, \text{kg}}{65 \, \text{kg/passenger}} \approx 28.24 \] Therefore, the maximum integer number of passengers that can safely ride is 28.

Key Concepts

Power Calculations

Understanding power is crucial when solving physics problems involving machines like elevators. Power is defined as the rate at which work is done. In our exercise, the power needed to lift an elevator is calculated using the equation:

• \[P = \frac{W}{t}\]

The work done, \(W\), depends on the force exerted and the distance covered. The force is the weight of the elevator, which is the product of mass and gravitational acceleration \( (F = m \cdot g) \). This means work is calculated as:

• \[W = m \cdot g \cdot d\]

Power calculations help us determine if the elevator's motor can handle the lifting task by comparing the power needed with the motor's power capabilities. Here, power conversion is necessary because motor power is given in horsepower, while calculations use watts, requiring a conversion factor of 746 W/hp.

Elevator Mechanics

Elevator mechanics often focus on understanding how various forces act on the elevator and how these influence its operation. Key insights from our exercise include:

• The elevator has a specific mass, which contributes to the force that needs to be counteracted for it to move upward.
• By ascending at a steady, constant speed, the elevator overcomes gravitational force without needing to accelerate further. Thus, the power used is purely to counteract gravity over the distance it moves.
• For safety and efficiency, motors are designed to provide enough power to lift both the elevator and any additional load (i.e., passengers) within a specified time limit. Here, an elevator moves five floors in 16 seconds at a steady speed, reflecting these considerations.

The design of elevators considers both these mechanical principles and operational power dynamics to ensure smooth and safe ascension and descension.

Work and Energy

Work and energy are fundamental concepts in physics, especially when analyzing machines like elevators. Work, in physics, is done when a force causes an object to move. The energy used to perform this work is stored and transferred in form of potential and kinetic energy.

• In the case of an elevator, potential energy increases as the elevator is lifted vertically against the force of gravity.
• The total work applied equates to the change in potential energy as the elevator raises to a higher floor.
• Energy efficiency is significant in these calculations, as it determines how well the elevator converts motor power into the lifting work required.

Understanding these principles helps us appreciate why certain amounts of power are required for lifting a specific mass and why energy conservation is essential in mechanical systems.